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M9/10_5-1 43675
MathematicsStage 5
MS5.1.1 Perimeter and area
Number: 43675 Title: MS5.1.1. Perimeter and Area
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Extract from Mathematics Syllabus Years 7-10 ©Board of Studies, NSW 2002 Unit overview pp iii, iv
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Unit overview i
Unit contents
Unit overview ....................................................................................... iii
Outcomes ................................................................................. iii
Indicative time............................................................................v
Resources..................................................................................v
Icons .........................................................................................vi
Glossary................................................................................... vii
Part 1 Straight-sided figures ............................................... 1–100
Part 2 Figures with curved sides......................................... 1–60
Unit evaluation ...................................................................................61
ii MS5.1.1 Perimeter and area 03/05
Unit overview iii
Unit overview
In this unit you will explore perimeter and areas of a variety of shapes.
You will look for common links between areas of figures and develop
formulas to describe some areas.
You will be given the opportunity to apply your understanding of
perimeter and area to practical situations, exploring ways of dividing
shapes into familiar two-dimensional figures. You will integrate your
geometry skills with your knowledge of measurement and units.
OutcomesBy completing the activities and exercises in this unit, you are working
towards achieving the following outcomes.
You have the opportunity to learn to:
• developing and using formulae to find the area of quadrilaterals:
– for a kite or rhombus, Area = � 12xy where x and y are the lengths
of the diagonals;
– for a trapezium, Area =� 12h(a + b) where h is the perpendicular
height and a and b the lengths of the parallel sides
• calculating the area of simple composite figures consisting of two
shapes including quadrants and semicircles
• calculating the perimeter of simple composite figures consisting of
two shapes including quadrants and semicircles.
You have the opportunity to learn to:
• identify the perpendicular height of a trapezium in different
orientations (Communicating)
• select and use the appropriate formula to calculate the area of a
quadrilateral (Applying Strategies)
iv MS5.1.1 Perimeter and area 03/05
• dissect composite shapes into simpler shapes (Applying Strategies)
• solve practical problems involving area of quadrilaterals and simple
composite figures (Applying Strategies).Source: Extracts from outcomes of the Maths Years 7–10 syllabus
<www.boardofstudies.nsw.edu.au/writing_briefs/mathematics/mathematics_710_syllabus.pdf > (accessed 04 November 2003).© Board of Studies NSW, 2002.
Unit overview v
Indicative timeThis unit has been written to take approximately 8 hours.
Each part should take approximately 4 hours.
Your teacher may suggest a different way to organise your time as you
move through the unit.
ResourcesResources used in this unit are:
For part 1 you will need:
• match sticks or toothpicks
• Internet
• a scientific calculator
• a ruler
• string or wool
• scissors.
For part 2 you will need:
• Internet
• a scientific calculator
• a ruler
• a pair of compasses.
vi MS5.1.1 Perimeter and area 03/05
IconsHere is an explanation of the icons used in this unit.
Write a response or responses as part of an activity.
An answer is provided so that you can check your
progress.
Compare your response for an activity with the one in the
suggested answers section.
Complete an exercise in the exercises section that will be
returned to your teacher.
Think about a question or problem then work through the
answer or solution provided.
Access the Internet to complete a task or to look at
suggested websites. If you do not have access to the
Internet, contact your teacher for advice.
Unit overview vii
GlossaryThe following words, listed here with their meanings, are found in the
learning material in this unit. They appear in bold the first time they
occur in the learning material. For these words and their meanings
including pronunciation see the online glossary on the CLI website at
http://www.cli.edu.au/Kto12 and follow the links to Stage 5 mathematics.
adjacent Another word for ‘next to’.
These two sides are adjacent.
algebra A branch of mathematics that uses letters to representnumbers or concepts.
arc Part of a circle or curve.
area The amount of surface. It is usually measured in squareunits, such as square centimetres, square millimetres,square metres, or square kilometres.
bisect To cut exactly in half.
circumference The distance around the outside of a circle. The namegiven to the perimeter of a circle.
composite Another word for ‘more than one’. For example: acomposite shape is made up of several simpler shapes
diagonal A line drawn from one corner to another corner of aplane shape.
dimension Length, breadth, width, height, thickness, depth.These are all dimensions.
equilateral All sides of equal length. Often used to describetriangles that have three equal sides.
figure Another word for shape.
formula A rule or principle stated in the language ofmathematics. Often written using algebraic symbols.
isosceles At least two sides equal.
perimeter The distance around the boundary of a two-dimensionalshape.
perpendicular At right-angles to. Meeting at 90°.
plane A flat surface.
product The result of multiplying eg the product of 2 and 3 is 6.
quadrant A quarter of a circle.
viii MS5.1.1 Perimeter and area 03/05
quadrilateral A plane shape with four straight sides.
rhombus A quadrilateral with four equal sides.
semicirculararc
Half the circumference of a circle.
substitute To replace one thing with another. Insertion of aninteger or number for a pronumeral.
tessellate To cover a plane surface with repeated shapes leavingno gaps and with no overlaps.
Mathematics Stage 5
MS5.1.1 Perimeter and area
Part 1 Straight sided figures
Part 1 Straight sided figures 1
Contents – Part 1
Introduction – Part 1..........................................................3
Indicators ...................................................................................3
Preliminary quiz – Part 1 ...................................................5
Area and perimeter..........................................................11
Simple composite figures ................................................17
Triangles and parallelograms..........................................25
Parallelograms.........................................................................30
Area of a rhombus and a kite ..........................................33
Kites .........................................................................................37
Area of a trapezium.........................................................41
Area of composite figures................................................47
Suggested answers – Part 1 ...........................................51
Additional resources – Part 1 ..........................................69
Exercises – Part 1 ...........................................................77
2 MS5.1.1 Perimeter and area 03/05
Part 1 Straight sided figures 3
Introduction – Part 1
In this part you will review the perimeter and area of some basic
quadrilaterals and the triangle. You will also develop the formulas for
the area of a rhombus and a trapezium. This will enable you to find the
perimeter and area of simple composite figures consisting of two shapes.
Indicators
By the end of Part 1, you will have been given the opportunity to work
towards aspects of knowledge and skills including:
• developing and using the formula for the area of a kite and rhombus
• developing and using the formula for the area of a trapezium
• calculating the perimeter and area of simple composite figures
consisting of two shapes that have no curved sides.
By the end of Part 1, you will have been given the opportunity to work
mathematically by:
• identifying the perpendicular height of a trapezium in different
orientations
• selecting and using the appropriate formula to calculate the area of a
quadrilateral
• dissecting composite shapes into simpler shapes
• solving practical problems involving area of quadrilaterals and
simple composite figures, with straight sides.
4 MS5.1.1 Perimeter and area 03/05
Part 1 Straight sided figures 5
Preliminary quiz – Part 1
Before you start this part, use this preliminary quiz to revise some skills
you will need.
Activity – Preliminary quiz
Try these.
1 Write the best name for these quadrilaterals, by using the markings
on their sides and in their angles to help you.
a
_______________________
b
_______________________
c
_______________________
d
_______________________
6 MS5.1.1 Perimeter and area 03/05
2 Each of the formulas can be used to determine the area of a type of
figure. Match the area formula to the correct figure, by drawing a
line to link them.
A = length × breadth
A =1
2of the base × height
A = length of side2
3 A square is 5 metres long.
a Calculate its area.
___________________________________________________
___________________________________________________
b Calculate its perimeter.
___________________________________________________
___________________________________________________
Part 1 Straight sided figures 7
4 The rectangle below is 6 metres long and 4 metres wide.
It is not drawn to scale.
6 m
4 m
a Calculate its area
___________________________________________________
___________________________________________________
b Draw a sketch of another rectangle on the grid below that has
the same area but different dimensions. Write the dimensions
on your diagram in the correct place.
What made you choose these dimensions?__________________
___________________________________________________
___________________________________________________
8 MS5.1.1 Perimeter and area 03/05
5 a Calculate the perimeter of the figure below.
8 m
2 m
___________________________________________________
___________________________________________________
b Draw a sketch of another rectangle in the space below that has
the same perimeter as the figure above. Write dimensions on
your figure.
6 Calculate the area of a triangle that has a base of 8 centimetres and a
perpendicular height of 3 centimetres.
_______________________________________________________
_______________________________________________________
_______________________________________________________
7 Complete the following statements that describe what is true about
each shape. (These statements are called properties.) Use the word
list: perpendicular, right angles, diagonals.
a The diagonals of a square meet at ________________________
b The ________________ of a rhombus meet at right angles.
c The diagonals of a kite are ______________________ to
each other.
Part 1 Straight sided figures 9
8 Explain what is meant by the statement ‘The diagonals of a square
and rhombus bisect each other but the diagonals of a kite do not.’
(You may use diagrams to assist your explanation.)
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
10 MS5.1.1 Perimeter and area 03/05
Part 1 Straight sided figures 11
Area and perimeter
Use a blue pencil to colour the area of the shape below.
Use a red pencil to show its perimeter.
Did you colour it correctly? You should have blue shaded inside the
shape and there should be a thin line of red around the outside of the
blue area.
Remember that area is the amount of space inside a plane (or flat) shape
and perimeter is the total length around the outside of that shape.
The figure in the activity below has no numbers. You will need to count.
12 MS5.1.1 Perimeter and area 03/05
Activity – Area and perimeter
Try these.
1 Use the diagram below to answer the following questions.
a Calculate the area of this rectangle.
___________________________________________________
___________________________________________________
b Calculate the perimeter.
___________________________________________________
___________________________________________________
Check your response by going to the suggested answers section.
You can calculate area by either counting the little squares inside a shape
or you can use the formula for the area of that shape.
You can find the perimeter of a shape by either adding up the total of all
the lengths around the outside of the shape or by recognising that because
there are some equal sides in your shape, you can multiply as well as add.
Part 1 Straight sided figures 13
But I can’t remember which is area andwhich is perimeter?
The last part of the word perimeter lookslike the word metre. A metre is a length, justlike perimeter.
In the next activity you will investigate figures that have the same
perimeter and figures that have the same area.
Activity – Area and perimeter
Try these.
2 A square is 3 cm long.
a Draw an accurate diagram of this shape. Write the length of
each side on your diagram.
b Calculate the perimeter of the square.
Write the correct units next to your answer.
___________________________________________________
___________________________________________________
c Calculate the area of this square, writing the correct units next to
your answer.
___________________________________________________
___________________________________________________
14 MS5.1.1 Perimeter and area 03/05
d Draw a rectangle that has the same area as the square above.
e Is the perimeter of your rectangle the same as the perimeter of
your square? _________________________________________
Explain _____________________________________________
___________________________________________________
f Draw a rectangle that has the same perimeter as the
square above.
g Is the area of your second rectangle the same as the area of the
square above? ________________________________________
Explain _____________________________________________
___________________________________________________
Check your response by going to the suggested answers section.
Shapes that have the same area can have different perimeters.
Part 1 Straight sided figures 15
Also, shapes with the same perimeter can have different areas.
A good way of understanding this last idea is to tie two ends of a piece of
string together. This gives you a fixed perimeter. Put the string around
four of your fingers. Move your fingers together first and then apart,
like this.
Thin rectangle
Wider rectangle
The thinner rectangle has a smaller area than the wider one.
Make your string into any shape you like. All these shapes have the
same perimeter.
Use the exercise below to make sure you understand the difference
between area and perimeter.
Go to the exercises section and complete Exercise 1.1 – Area and
perimeter.
16 MS5.1.1 Perimeter and area 03/05
Part 1 Straight sided figures 17
Simple composite figures
Composite plane figures can have two or more shapes glued together to
make a new shape.
You already know that different areas can have the same perimeter.
You are going investigate this further using tessellated (repeated) shapes
that create composite figures.
The series of pictures below show the steps used to build up a shape.
figure 1 figure 2 figure 3
figure 4
figure 5
The final diagram contains five repeated rhombuses. Each rhombus has
the same area and the same perimeter.
The area of each rhombus is 0.9 cm2 .
1 cm
Area = 0.9 cm2.
Use the diagrams above when doing the following activity.
18 MS5.1.1 Perimeter and area 03/05
Activity –Simple composite figures
Try these.
1 a Use figure 5, above, to complete the last row of this table.
Number ofrhombuses
Area in cm2 Perimeter in cm
1 0.9 4
2 1.8 6
3 2.7 8
4 3.6 10
5
b Another rhombus is added to figure 5, above, like this:
Use this new shape to complete the first row of the table below.
Number ofrhombuses
Area in cm2 Perimeter in cm
6
7
8
c Add the seventh and eighth rhombus next to the shaded rhombus
in the picture above. Use these to complete the table.
d What do you notice about the perimeter of the shape, even
though you are increasing its area.
___________________________________________________
Part 1 Straight sided figures 19
Check your response by going to the suggested answers section.
The perimeter of some shapes remains the same even when you cut bits
out of them. For example, the following two shapes both have the
same perimeter.
8 cm
5 cm
8 cm
5 cm
The opposite sides of the first shape are the same. In the second shape,
the two sides opposite the 5 cm side have a total of 5 cm. Perhaps these
two sides may be 1 cm and 4 cm, or 2 cm and 3 cm, or any other pair of
numbers that add to 5 cm. You can also see that the two sides opposite
the 8 cm side add to 8 cm.
There are many lengths that these four unknown sides may be.
Below are two possibilities. However, the total of each part of one side
is always equal to the length of its opposite side.
8 cm
5 cm
4 cm
1 cm
3 cm
5 cm
8 cm
5 cm
3 cm
2 cm
1 cm
7 cm
Go to the two web activities called Magical Matches by visiting the CLI
webpage below. Select Mathematics then Stage 5.1 and follow the links
to resources for this unit MS5.1.1 Perimeter and Area Part 1.
<http://www.cli.nsw.edu.au/Kto12>
20 MS5.1.1 Perimeter and area 03/05
Below is the calculation for the perimeter of a rhombus and a right-
angled triangle.
1 cm
Perimeter = 5 + 5 + 5 + 5 or
= 5 × 4 (because all the sides are the same)
= 20 cm
5 cm
4 cm
3 cm
Perimeter = 4 + 3 + 5
=12 cm
(Remember, calculate perimeter the same way every
time. Start at the top of the triangle and move
around clockwise.)
The class teacher drew this rhombus and right-angled triangle together.
The new shape looked like this:
start here5 cm
4 cm
3 cm
What is the perimeter ofthe new shape?
32 cm. Just add 20 and 12.
No. The 5 cm side of the triangle isnow not part of the new perimeter.
And neither is oneside of the rhombus!
Perimeter = 5 + 4 + 3 + 5 + 5
= 22 cm
Part 1 Straight sided figures 21
(For a quicker calculation, you could triple the five and then add the four
and three.)
If you are not getting the correct answers, try starting at the top left hand
corner of the shape and then move clockwise around the shape adding the
lengths as you go. Do this for all composite figures, every time.
Activity – Simple composite figures
Try these.
Answer the following questions about the composite figures below.
These figures have not been drawn to scale.
2 This figure contains a rectangle and a right-angled triangle.
x cm
8 cm
10 cm
20 cm 6 cm
a How long is x? ______________________________________
b Calculate the perimeter. _______________________________
___________________________________________________
22 MS5.1.1 Perimeter and area 03/05
3 This figure has a parallelogram sitting on top of a rectangle.
5 mm
6 m
m
8 mm
a Write down the missing lengths on the diagram above.
b Calculate the perimeter. _______________________________
___________________________________________________
4 All the corners in this figure are right angles.
5 m
2 m
9 m
a The bottom of this figure has no length written on it.
Write its length on the diagram.
b What is the sum of the lengths of the two sides opposite the
9 m side? ___________________________________________
c Choose appropriate lengths for these two sides and write them
on the diagram above. (There are many possible answers.)
d What is the perimeter of this figure? _____________________
___________________________________________________
___________________________________________________
Check your response by going to the suggested answers section.
Part 1 Straight sided figures 23
Before you find the perimeter of a plane (flat) figure, you need to make
sure that you know all the lengths around the outside.
Remember, always look for sides that are equal in length such as the
opposite sides of a rectangle or the adjacent sides (sides next to each
other) of a square.
You have been increasing area and finding that perimeter can remain the
same. You have also been practising finding the perimeter of
composite figures. Now check that you can solve these kinds of
problems by yourself.
Go to the exercises section and complete Exercise 1.2 – Simple composite
figures.
24 MS5.1.1 Perimeter and area 03/05
Part 1 Straight sided figures 25
Triangles and parallelograms
Triangles have three sides. The base is not necessarily on the bottom of
the triangle. However, the height is always perpendicular (at 900 ) to its
base, as in the diagrams below.
height
base
height
bas
e
Remember:
Area of triangle =1
2× base × perpendicular height
or A =1
2bh
Explore why this formula works in the following activity.
26 MS5.1.1 Perimeter and area 03/05
Activity – Triangles and parallelograms
Try these.
1 Answer the following questions about the diagram below.
A
B
a The length of the rectangle is 5 units.
How wide is the rectangle?
___________________________________________________
b Calculate the area of the rectangle. ______________________
___________________________________________________
c i Draw a line from A to B so you divide your rectangle and
triangle into two parts. Compare the shaded section of each
part with the unshaded section. What do you notice?
________________________________________________
ii Complete this sentence.
The area of the triangle is _______________ the area of the
rectangle.
d Use the fact that the area of a triangle = 1
2× area of rectangle to
calculate the area of the triangle above.
___________________________________________________
Part 1 Straight sided figures 27
e Complete these sentences.
i The width of the rectangle is the same as the ____________
of the triangle.
ii The length of the rectangle is the same as the ___________
of the triangle.
iii To find the area of a triangle you must ______________ the
area of the rectangle. To do this you must multiply the
length of the rectangle by the _____________ and
___________ by two. This is the same as multiplying the
length of the triangle’s ____________ by its height and
dividing by two.
f Complete for the figure above: If b = ____ and h = ______ ,
then
A = ___ × b × h
= ___× 4 × 5
= 10 square units
Check your response by going to the suggested answers section.
To calculate the area of any triangle, you can:
• calculate the area of the rectangle surrounding it, then halve this
answer to find the area of the triangle
or
• use the formula for the area of a triangle, A =1
2bh .
The activity below, shows you different ways of thinking about this
area formula.
28 MS5.1.1 Perimeter and area 03/05
Activity – Triangles and parallelograms
Try these.
2 The base of a triangle is 8 metres long and its perpendicular height is
12 metres. By doing the calculation inside the brackets first, show
that the following three expressions give the same answer for the
area of the triangle.
a1
2× (b × h) __________________________________________
___________________________________________________
b (1
2× b) × h __________________________________________
___________________________________________________
c (1
2× h) × b __________________________________________
___________________________________________________
3 There are four lengths given to you in the triangle below.
9 m7.6 m
7 m
3 m
a How long is the:
i base ____________________________________________
ii perpendicular height _______________________________
Part 1 Straight sided figures 29
b Use A =1
2bh to calculate the area of this triangle.
___________________________________________________
___________________________________________________
___________________________________________________
c What are the two lengths you did not need when you calculated
the area of this triangle? ________________________________
Check your response by going to the suggested answers section.
When calculating the area of any triangle, using A =1
2bh make sure you
use its base and perpendicular height and no other lengths.
Also, try to choose the easiest way to do your calculations.
• If the base is an even number, then halve the base.
• If the perpendicular height is an even number, then halve the height.
• If neither dimension is an even number, then divide your answer to
(base × height) by two.
• If both dimensions are even numbers, just remember to halve only
one number.
30 MS5.1.1 Perimeter and area 03/05
Parallelograms
The class teacher writes the following formula for the area of a
parallelogram on the board.
Area = length of parallelogram × perpendicular height
She then hands out the following diagram.
base length
perpendicular height
Use a pair of scissors to show why this isthe correct way to calculate the areaof a parallelogram.
Use scissors to cut the parallelogram provided for you in
additional resources.
John shows the class that if you cut the parallelogram along the dotted
perpendicular height, you get a right-angled triangle. This triangle can
be moved to the other side of the parallelogram to form a rectangle.
length of parallelogram
perpendicularheight
The area of this rectangle must be the same as the area of
the parallelogram.
Part 1 Straight sided figures 31
Use the diagrams and information above to answer the questions in the
next activity.
Activity – Triangles and parallelograms
Try these.
4 Complete these sentences:
a The width of the rectangle is the same as the _______________
of the parallelogram.
b The length of the rectangle is the same as the _______________
of the parallelogram and its opposite side.
5 One pair of parallel sides of a parallelogram measure 10 cm.
The height perpendicular to these sides is 7 cm long.
a Write these dimensions on the parallelogram below.
b Calculate the area of the parallelogram.
___________________________________________________
___________________________________________________
___________________________________________________
32 MS5.1.1 Perimeter and area 03/05
6 Explain why you cannot do this area calculation for the
parallelogram below.
7 m
m
3 mm
Area of parallelogram = length × height
= 3 × 7
= 21 mm2
_______________________________________________________
_______________________________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
If you are given the sides of a parallelogram, you can find the perimeter
but you cannot find its area. You can only find its area if one dimension
is perpendicular to the other. The four sides of a parallelogram are not
perpendicular. (If they are, then you call it a rectangle.)
You have been revising finding the area of triangles and parallelograms.
The following exercise also reviews finding areas of squares
and rectangles.
Go to the exercises section and complete Exercise 1.3 – Triangles and
parallelograms.
Part 1 Straight sided figures 33
Area of a rhombus and a kite
A rhombus has four equal sides. If you push a square it becomes a
rhombus. (This is because a square is just a special type of rhombus.)
Apart from having four equal sides, the diagonals inside a rhombus and
a square meet at 900 . These diagonals also bisect each other.
(This means that they cut each other in half.)
When you push the square, one diagonal gets longer and the other gets
shorter, but the two diagonals always remain at 900 , and always cut each
other in half.
You will investigate the area of a rhombus in the next activity.
Use the rhombus provided for you in the additional resources.
This rhombus has a rectangle around the outside of it.
34 MS5.1.1 Perimeter and area 03/05
Activity – Area of a rhombus and a kite
Try these.
1 Fold the rhombus along the dotted line and look at only half of it.
a Describe what you are looking at by naming shapes and stating
their position.
___________________________________________________
___________________________________________________
b What area is the shaded triangle compared to the rectangle?
___________________________________________________
c Open your rhombus. Both halves of your diagram are the same.
Complete this sentence: The area of the rhombus is
____________ the area of the rectangle that surrounds it.
2 a Your rhombus and rectangle are on grid paper.
If your rectangle is 14 metres long, what does each square on the
grid paper represent? _________________________________
b How wide is your rectangle? ___________________________
c Calculate the area of the rectangle.
___________________________________________________
___________________________________________________
d Show how you can now calculate the area of the rhombus.
(Show your working. Don’t just give the answer.)
___________________________________________________
___________________________________________________
Part 1 Straight sided figures 35
3 a i How long are the two diagonals of the rhombus?
________________________________________________
ii Complete this sentence.
The two __________________ of the rhombus are the same
length as the ______________ and ________________ of
the rectangle that surrounds it.
b i Multiply the lengths of these two diagonals together.
________________________________________________
ii Halve this answer. ________________________________
What do you notice? ______________________________
________________________________________________
iii Complete this sentence.
The product of the two diagonals is the same as the area
________________________________________________
iv Complete these sentences.
Area of rhombus = 1
2 of the ______________ of its two
diagonals.
Area of rhombus = ___ × ___________× other diagonal.
Check your response by going to the suggested answers section.
36 MS5.1.1 Perimeter and area 03/05
You can write area formulas in a shorter form using algebra.
The last formula is a shorter way of writing the area of a rhombus.
You know the first four already.
l
l
Area of square = l × l
= l2
b
l
Area of rectangle = lb
h
b
Area of triangle = 12bh
h
l
Area of parallelogram = lh
x
y
Area of rhombus = 12xy
Part 1 Straight sided figures 37
There is one thing that all the dimensions above have in common.
What is it?
Look at the diagrams above. Both dimensions of each shape are
perpendicular (at 900 ) to each other. You can only multiply
perpendicular lengths together to get area.
Kites
Imagine that two of the sides of your rhombus are made of elastic and
that you could stretch them like this.
What shape is it now?
It’s got two different isosceles trianglesin it.
It’s a kite.
38 MS5.1.1 Perimeter and area 03/05
Look carefully at the area of the new triangle.
126
12
The area of this rectangle above is 72 m2 (6×12 ) and the area of the
triangle is 12
×12 × 6 = 12
of 72. So the area of the triangle is half the
area of the rectangle, just like before.
How does the area of the kite compare to the whole new rectangle?
It’s just like the rhombus. The kite is halfthe new rectangle.
This means that the formula for a kite is the same as a rhombus.
x
y
Area of kite = 12xy
Part 1 Straight sided figures 39
The class checks that this is right, by using the formulas above.
Rectangle around kite has l = 19, b = 6
Area of new rectangle = lb
= 19 × 6
= 114 m2
Diagonals of kite are x = 6 and y = 19 (or x = 19 and y = 6)
Area of kite =1
2xy
=1
2× 6 × 19
= 3 × 19 (by halving 6)
= 57 m2
57 is half of 114, so it’s true.
Use the formula for the area of a kite in the following activity.
Activity – Area of a rhombus and a kite
Try these.
4 Complete the following using the kite below.
8 cm
25
cm
a Copy the lengths of the two diagonals below:
x = ________________________________________________
y = ________________________________________________
40 MS5.1.1 Perimeter and area 03/05
b Substitute these two numbers into the formula, A =1
2xy and
then calculate the area of the kite.
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
Check your response by going to the suggested answers section.
Remember that this formula can also be used for rhombuses.
To calculate areas of both kites and rhombuses, you only need the
lengths of the two diagonals. Also, remember to halve only once when
doing your calculations.
Use the formula A =1
2xy in the following exercise.
Go to the exercises section and complete Exercise 1.4 – Area of a rhombus
and a kite.
Part 1 Straight sided figures 41
Area of a trapezium
A trapezium has four sides and at least one pair of parallel sides.
It may look like this:
Of course most people call the last figure a rectangle, because rectangles
are special kinds of trapeziums.
You will investigate the area of a trapezium in the next activity.
Use the two trapeziums provided for you in the additional resources.
Activity – Area of a trapezium
Try these.
1 Cut out only one of these trapeziums and place it on top of the other.
a Are both these trapeziums the same size? __________________
b Do they have the same area? ____________________________
c Place the cut trapezium next to the other trapezium, so that you
form a large parallelogram. (You will need to rotate your cut
trapezium.)
d Carefully trace around three of the sides of your cut trapezium
so you can clearly see the parallelogram you have formed.
Remove your cut trapezium.
42 MS5.1.1 Perimeter and area 03/05
e Complete this sentence.
The area of the trapezium is _______________ the area of the
parallelogram.
f i Imagine that each square on your grid paper is one metre by
one metre, so that each square has an area of 1 m2 .
(If this is difficult to do, try imagining that you are on the
top of a building looking down at these square metres on the
grass below. From that height they would look much
smaller than metres.)
ii Complete these sentences.
Length of parallelogram, l = ________________________
Height of parallelogram, h = ________________________
Area of parallelogram = l __
= __ × __
= 70 ___
g Show how you can calculate the area of the trapezium.
___________________________________________________
___________________________________________________
2 Look at only one trapezium.
a How far apart are its parallel sides? _____________________ .
b Choose the best words from the list below and write them in the
sentences:
from between parallelogram trapezium
The height of the parallelogram is the same as the distance
_________________ the parallel sides of the trapezium.
This distance is also the height of the ____________________ .
Part 1 Straight sided figures 43
c i How long is the shorter parallel side? _________________
ii How long is the longer parallel side? __________________
iii Add the lengths of the two parallel sides together.
________________________________________________
iv Complete this sentence.
The sum of the two _________________ sides in the
trapezium = the _______________ of the parallelogram.
v Complete this sentence.
Area of trapezium =1
2 × ______ of the ___________ sides × height .
vi Show how you can find the area of this trapezium by first
finding the sum of the parallel sides.
________________________________________________
________________________________________________
________________________________________________
________________________________________________
Check your response by going to the suggested answers section.
To find the area of a trapezium you can:
1 Add the lengths of the two
parallel sides together.
This is the same as the length of the
parallelogram.
2 Multiply this sum by the
perpendicular height of the
trapezium.
The perpendicular height of the
trapezium is the distance between
the two parallel sides and the same
as the height of the parallelogram.
3 Halve your answer The area of the trapezium is half the
area of the parallelogram.
44 MS5.1.1 Perimeter and area 03/05
So area of trapezium =1
2 the sum of the parallel sides × height .
The class try this with the trapezium below.
12 cm
7 cm
8 c
m
12 + 7 = 19
1
2of 152 is 76 cm2
press 151 ÷ 2 = .5 cm2Just use your calculator and
What if it’s an oddnumber, like 151?
19 × 8 = 152
75
Mathematicians use algebra to write down formulas. If you call the
parallel sides a and b, and the height of the trapezium h, then you get a
picture like this:
b
a
h
(Remember that h is always 900 to the parallel sides.)
Part 1 Straight sided figures 45
The formula that goes with this diagram is:
Area of trapezium =1
2× (a + b) × h or
A =1
2h(a + b)
Use the methods shown above in the next activity.
Activity – Area of trapezium
Try these.
3 Calculate the area of the trapezium below.
7 m
m
10 m
m
5 mm
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
46 MS5.1.1 Perimeter and area 03/05
4 If a and b are the lengths of the parallel sides of a trapezium and h is
its perpendicular height, calculate the area of this trapezium, using
the information below.
a = 9 cm
b = 5 cm
h = 10 cm
A =1
2h(a + b)
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
Using the formula is often faster because you can choose to halve the
larger even number.
If you use your calculator you must remember to add the lengths of the
parallel sides together first (and press “equals” to get the answer.)
Alternatively, you could use brackets, remembering to use ( as well as
) . You can also use the fraction key ab⁄c instead of dividing by two.
Use your calculator in the question above. Can you get the same answer?
You have been practising finding the area of trapezium.
Now check that you can solve these kinds of problems by yourself.
Go to the exercises section and complete Exercise 1.5 – Area of a
trapezium.
Part 1 Straight sided figures 47
Area of composite figures
Composite figures can have more than one shape inside them.
They also may have shapes cut out of them. Here are two examples.
The composite areas are shaded.
To calculate the area of the first figure, you must add the triangular area
and trapezium area together. To calculate the area of the second figure
you must subtract the smaller rectangular area from the larger
rectangular area.
Some figures can be divided different ways.
Can you think of the first figure above in a different way?
48 MS5.1.1 Perimeter and area 03/05
There are many different shapes in composite figures.
However, when calculating area it is always best to think carefully about
what shapes you are going to choose. Choosing the smallest number of
shapes means you have less calculations to do.
Activity – Area of composite figures
Try these.
1 Answer the following questions about the figure below.
2 m 2 m
5 m 5 m
4 m
The rectangle above has a triangle cut out of it.
Draw a dotted line along the base of the triangle so you can see it
more clearly.
a How long are the base and height of this triangle?
___________________________________________________
Indicate exactly where these dimensions are, on the
diagram above.
Part 1 Straight sided figures 49
b Calculate the area of the triangle.
___________________________________________________
___________________________________________________
___________________________________________________
c What are the dimensions of the rectangle? __________________
d Calculate the area of the rectangle.
___________________________________________________
___________________________________________________
e Calculate the area of the shaded composite figure by subtracting
one area from the other.
___________________________________________________
2 A farm property was surveyed along its length from corner to corner.
The diagram below shows the results of the survey.
2750 m
2900 m
630
0 m
2500
m
(This is called a traverse survey.)
a There are many shapes inside this figure. However, there are
two that make up the whole of this property. Write the names of
the two shapes.
___________________________________________________
50 MS5.1.1 Perimeter and area 03/05
b Calculate the area of the property by finding the area of these
two shapes first, then adding these areas together.
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
Check your response by going to the suggested answers section.
Areas of composite shapes are often needed in many aspects of real life.
For example, in the building industry, area needs to be calculated for
such things as painting, laying of bricks, or ordering decking materials.
Area is also needed in land management. For example, after a bushfire,
to stop soil from being blown or washed away, seed is sometimes
scattered by aeroplanes. You need to know the area of land the seed is
covering in order to plan the amount of seed to purchase.
Use your knowledge of the areas of composite shapes to complete the
exercise below.
Go to the exercises section and complete Exercise 1.6 – Area of composite
figures.
Part 1 Straight sided figures 51
Suggested answers – Part 1
Check your responses to the preliminary quiz and activities against these
suggested answers. Your answers should be similar. If your answers are
very different or if you do not understand an answer, contact your teacher.
Activity – Preliminary quiz
1 a square. All the sides are equal and each vertex (corner) is a
right angle.
b rhombus. All the sides are equal.
c trapezium. At least one pair of opposite sides are parallel.
(The two right angles are not an important feature of a
trapezium. These angles just mean that the two parallel sides
both meet another side at 900 .)
d kite.
A kite is the shape that flies in the sky.
It has two different isosceles triangles in it.
52 MS5.1.1 Perimeter and area 03/05
2
A = length x breadth
A = 12
of base x height
A = length of side 2
3 a 25 m2 .
Area = length of side2
= 52
= 5 × 5
= 25 m2
b 20 m.
5 m5 m
5 m
5 m
start here
Starting from the top left corner and going clockwise, add
5 +5 + 5+ 5 = 20. Alternatively, you will notice that all the
sides are equal, so you can simply calculate 4 × 5 = 20 .
Part 1 Straight sided figures 53
4 a 24 m2
Area = length × breadth
= 6 × 4
= 24 m2
b There are many rectangles you can draw. Here are two.
8 m
3 m
12 m
2 m
(If you are not sure if your diagram is correct, contact
your teacher.)
These dimensions were chosen because they multiply together
to give the same area of 24 m2 .
5 a 20 m. Again, starting at the top left of your diagram and going
clockwise, Perimeter = 8 + 2 + 8 + 2 = 20 m. Alternatively, you
will notice that the opposite sides have the same length, so:
Perimeter = 2 × (8+2)
= 2 × 10
= 20 m
b There are many rectangles you can draw. Here are two.
9 m
1 m
6 m
4 m
One length and one breadth must add to a total of 10 m.
(If you are not sure whether your diagram is correct, contact
your teacher.)
6 12 cm2 .
Area =1
2 of base × height
=1
2 of 8 × 3
= 4 × 3 (by halving 8)
= 12 cm2
54 MS5.1.1 Perimeter and area 03/05
7 a right angles b diagonals c perpendicular
8 This means that in a square and a rhombus, each diagonal cuts the
other diagonal on half. However in a kite, only one diagonal cuts
the other diagonal in half, so you cannot use the words ‘each other’
when talking about the diagonals of a kite.
For example: if the diagonals of a square are 6 cm long then,
3 cm each
If the diagonals of a rhombus are 6 cm and 8 cm long then,
3 cm each
4 cm each
If the diagonals of this kite below are 4 cm and 10 cm, then
2 cm each
The other diagonal may be divided into lengths of 1 cm and 9 cm or
2 cm and 8 cm or 3 cm and 7 cm (or any two lengths that have a total
of 10 cm.)
Part 1 Straight sided figures 55
Activity – Area and perimeter
1 a Area = 35 square units (or 35 u2 )
Area is the total space inside the rectangle. There are 35 little
squares inside this rectangle. However, the answer is not just
35. You must write down the name of the units you are using.
For example: if you were in an aeroplane and looking down at a
large piece of land perhaps these units are square kilometres.
If this was under a microscope, perhaps they are even smaller
than square millimetres.
As there are no lengths in the diagram you can only write a
general term. That is 35 square units (or 35 u2 .)
Alternatively, because it is a rectangle, you can count the spaces
between the marks on two of the sides of the rectangle and get:
length = 7 units
width = 5 units
(Remember to use the word ‘units’ because you
do not know what each mark is.)
Area of rectangle = length × width
= 7 × 5
= 35 square units
b Perimeter = 7 + 5 + 7 + 5
= 24 units
If you use the this method, try to always start at the same place,
like the top left corner of the shape, counting the spaces between
the marks. Continue going around clockwise until you get back
to your starting position.
start here1 2 3
2324
This way you will never miss out a side.
56 MS5.1.1 Perimeter and area 03/05
Alternatively, you could do this:
perimeter = 2 × (length+width)
= 2 × (7+5)
= 2 × 12
= 24 units
This second method is faster. Both pairs of opposite sides are
the same. Once you know the length of one side, you know the
opposite side is the same.
2 a 3 cm
3 cm
3 cm
3 c
m
b Perimeter = 3+3+3+3
= 12 cm
or because all four sides are equal you can do this:
perimeter = 4 × 3
= 12 cm
c Area of square = 3 × 3 (or 32 )
= 9 cm2
d There are other more difficult rectangles you can draw.
This is the simplest:
9 cm
1 cm
(Another rectangle, for example, may be 0.9 cm by 10 cm.)
e No.
The rectangle above has a perimeter of 9 + 1 + 9 + 1 = 20 cm,
whereas the square has a perimeter of 12 cm.
Part 1 Straight sided figures 57
f There are many answers. Here are two:
4 cm
2 cm
2 cm
4 cm
5 cm
1 cm
5 cm
1 cm
Notice that one pair of adjacent sides add up to half the
perimeter. So to decide the length and width of your rectangle,
you must first divide the perimeter of your square by 2.
12 ÷ 2 = 6. This means that the length and width of your
rectangle must add up to 6, so the sides may be 5 and 1, or 4 and
2 or 3 and 3 (just like the square,) or 5.5 and 0.5. The list is
endless.
g No, the area of the rectangle is not 9 cm2 . It is different:
area of first rectangle, above = 2 × 4
= 8 cm2
area of second rectangle = 5 × 1
= 5 cm2
Activity – Simple composite figures
1 a5 4.5 12
Area = 0.9 × 5
= 4.5
(You may use a calculator if you wish or simply multiply 9 by 5
in your head and change it to 4.5. As there is one decimal place
in the question, there must be one decimal place in the answer.
This is because both question and answer are one tenth the size
of 9 × 5.)
Be careful when counting area. Always start at the top left of
the shape and count around clockwise.
58 MS5.1.1 Perimeter and area 03/05
b and c
Number ofrhombuses
Area in cm2 Perimeter in cm
6 5.4 12
7 6.3 12
8 7.2 12
d The perimeter of the shape remains the same, even though you
are increasing its area.
Isn’t that amazing!
2 a x = 14 cm.
The bottom of the rectangle = 20 − 6
= 14 cm
As the opposite sides of a rectangle are the same length, then the
top length is also 14 cm.
b Staring at the top left and moving around the shape in a
clockwise direction:
Perimeter = 14 + 10 + 20 + 8
= 52 cm
3 a
5 mm
6 m
m
8 mm
6 m
m
5 m
m
8 mm
The opposite sides of parallelograms and rectangles are all
equal.
b Starting at the top left again,
Perimeter = 8 + 6 + 5 + 8 + 5 + 6
= 2 × (8 + 6 + 5)
= 38 mm
Part 1 Straight sided figures 59
4 a 7 m. (5 + 2 = 7 m – see diagram below.)
In figures like this, always add the lengths that are parallel to the
side you are trying to find.
b 9 m. The two unknown sides must have the same total length as
the other side opposite them.
c You could choose 1 and 8 or 2 and 7 or 3 and 6 metres, etc.
You can write whatever you like so long as their sum is 9.
One possible set of lengths are:
5 m
2 m
9 m
1 m
8 m
7 m
d Perimeter = 5 + 1+ 2 + 8 + 7 + 9
= 32 m
Alternatively, its perimeter is twice the length plus width.
Perimeter = 2 × (9 + 7)
= 2 × 16
= 32 m
Activity – Triangles and parallelograms
1 a 4 units. (Remember to count the spaces, not the marks.)
b 20 square units.
You can draw lines in your rectangle, and then count the squares
or you can just multiply five by four because there are five rows
each with four squares in it.
Area = 5 × 4
= 20 square units
60 MS5.1.1 Perimeter and area 03/05
c i In each part, the shaded section is the same as the unshaded
section.
A
B
dia
gona
l
diagonal
Each part, on either side of AB, is divided into two sections
by a diagonal. In each part, both sides of each diagonal are
the same shape and have the same area.
ii half
d Area of triangle = 1
2× 20
= 10 square units
You can simply halve twenty or divide the two into the twenty.
e i base
ii height
iii halve, width, divide, base.
f b = 4 and h = 5, then
A = 1
2× b × h
=1
2× 4 × 5
= 10 square units
2 Make sure you substitute 8 into b and 12 into h, so that you get three
different ways of doing the same question.
a1
2× (bh) =
1
2× (8 ×12)
=1
2× 96
= 48 (by halving 96)
Part 1 Straight sided figures 61
b (1
2b) × h = (
1
2× 8) × 12
= 4 × 12 (by halving 8)
= 48
c (1
2h) × b = (
1
2× 12) × 8
= 6 × 8 (by halving 12)
= 48
3 a i 3m.
ii 7 m.
The base is always at right angles to its height. It is not always
on the bottom of the triangle. (Practise visualising this by
turning the triangle around so you can look at it in different
directions.
b 10.5 m2
A =1
2bh
=1
2× 3 × 7
=1
2× 21
Note that it as 3 and 7 are odd numbers, it is easier to simply
multiply them. To calculate the answer, you could either
mentally halve 21, or use a calculator like this:
• 1 ab⁄c 2 21 = or
• 21 ÷ 2 =
c 7.6 m and 9 m. Neither of these lengths are either the base or
the perpendicular height of the triangle.
4 a height b length (or base), opposite
5 a
10 cm
7 cm
62 MS5.1.1 Perimeter and area 03/05
b Area of parallelogram = length × height
= 10 × 7
= 70 cm2
6 The height of a parallelogram must be perpendicular to its length.
This is not true in the example given, so you cannot multiply three
by seven to get area. The two sides are not at right angles to each
other. You need the height in the diagrams below to find its area.
7 m
m
3 mm
height
7 m
m
3 mm
height
Activity – Area of a rhombus and a kite
1 a You are looking at an isosceles triangle (two sides equal) inside
a rectangle that surrounds it.
b Half. (Divide the shape into two parts, from the top of the
triangle (the apex) to the base of the triangle. Each of the two
shaded sections are the same size as the two unshaded sections.)
c half
2 a 1 m2 Each square is one metre by one metre and this equals one
square metre.
b 6 m. (Remember to count the spaces along the width.)
c Area of rectangle = length × width
= 14 × 6
= 84 m2
Alternatively, you could use
A = lb
= 14 × 6
= 84
Part 1 Straight sided figures 63
d Area of rhombus = 1
2× area of rectangle
= 1
2× 84
= 42 m2
3 a i 14 m and 6 m
ii diagonals, length, width (or breadth)
b i 14 × 6 = 84
ii1
2 of 84 = 42 . The answer is the same as halving the area
of the rectangle.
iii of the rectangle
iv product, 1
2, diagonal.
4 a x = 8 cm, y = 25 cm (or x = 25 cm, y = 8 cm because x and y are
used for both diagonals. )
b 100 cm2
A =1
2xy
=1
2× 8 × 25
Now halve the even number, 8. You get 4 .
= 4 × 25
= 100
Alternatively, after substituting, you could simply use a
calculator, like this: 1 ab⁄c 2 8 25 =
or this: 8 25 ÷ 2 =
64 MS5.1.1 Perimeter and area 03/05
Activity – Area of a trapezium
1 a Yes b Yes
c and d
e half
f i l = 14 m , h = 5 m .
ii Area of parallelogram = lh
= 14 × 5
= 70 m2
g Area of trapezium =1
2of area of parallelogram
=1
2× 70
= 35 m2
2 a 5 metres
count thisdistance
(“How far apart?” always means the shortest distance.
This distance is usually at 900 to at least one of the lines.)
b between, trapezium.
Part 1 Straight sided figures 65
c i 5 m ii 9 m
shorter parallel side
longer parallel side
(Remember, you must be careful to count the spaces.)
iii 5 + 9 = 14 metres.
iv parallel, length.
v sum, parallel.
vi Sum of parallel sides = 14 m and height of trapezium = 5 m.
Area of trapezium =1
2× 14 × 5
=1
2× 70
= 35 m2
(Notice that this is the same area as before when you halved
the area of the parallelogram.)
3 42.5 mm2 .
Sum of parallel sides = 7 + 10
= 17 mm
height = 5 mm
(Remember, the shortest distance
between the parallel sides is
the height.)
Area =1
2× 17 × 5
=1
2× 85
= 42.5 mm2
(Remember that halving a number
is the same as dividing it by two.
Using a calculator, you can press:
• 7 + 10 = × 5 ÷ 2 =. (You must press equals after adding.)
Alternatively, you can use brackets: ( 7 + 10 ) × 5 ÷ 2 =
• 1 ab⁄c 2 × ( 7 + 10 ) × 5 =
66 MS5.1.1 Perimeter and area 03/05
4 Always write the formula first, then substitute.
(Remove a letter and put a number in its place.)
A = 1
2h(a + b)
=1
2× 10 × (9 + 5)
=1
2× 10 × 14
At this point, you can choose to halve the 10 or halve the 14 or halve
the answer to 10 ×14 . If you are using your head, which is better?
Always try to reduce the size of your numbers, so halve 14, because
14 is the biggest number. (Halving 10 means you will get 5×14 and
this is more difficult to do in your head than when you halve 14.
Halving 14 means you need to only calculate 10 × 7 .
This is much easier.)
So A = 10 × 7 (by halving 14)
= 70
So area of trapezium is 70 cm2
Activity – Area of composite figures.
1 a base = 6 m, height = 4 m.
base = (5 + 5) − (2 + 2)
= 6 m
4 m
2m 2m
5 m 5 m
4 m
6 m
b 12 m2
Area of triangle = 1
2× base × height
= 1
2× 6 × 4
= 3 × 4 (by halving 6)
= 12 m2
Part 1 Straight sided figures 67
c 10 m by 4 m.
d 40 m2
Area = length × breadth
= 10 × 4
= 40 m2
e 28 m2
Shaded area = rectangular area − triangular area
= 40 − 12
= 28 m2
2 a trapezium, triangle
triangletrapezium
b 21 235 000 m2
Trapezium: h = 2750 m, a = 2500 m, b = 6300 m
Area of trapezium =1
2h(a + b)
= 1
2× 2750 × (2500 + 6300)
=1
2× 2750 × (8800)
= 2750 × 4400 (by halving 8800)
= 12 100 000 m2
Triangle: b = 6300 m, h = 2900 m
68 MS5.1.1 Perimeter and area 03/05
Area of triangle =1
2bh
=1
2× 6300 × 2900
= 3150 × 2900 (by halving the larger number)
= 9 135 000 m2
Total area = area of trapezium + area of triangle
= 12 100 000 + 9 135 000
= 21 235 000 m2
Part 1 Straight sided figures 69
Additional resources – Part 1
To be cut out and used in Activity – Triangles and parallelograms,
question 4.
70 MS5.1.1 Perimeter and area 03/05
Part 1 Straight sided figures 71
To be cut out and used in Activity – Area of a rhombus and a kite.
fold
her
e
72 MS5.1.1 Perimeter and area 03/05
Part 1 Straight sided figures 73
Cut out only one figure and use both pieces in Activity – Area of a
trapezium.
74 MS5.1.1 Perimeter and area 03/05
Part 1 Straight sided figures 75
Use this map in Exercise 1.6 – Area of composite figures.
Na
ttai
R
i v e r
Win
ge c a r r i b e e R i v e r
Shoa l h a v e n
Rive
r
Mul
war
eeR
iver
Wo
l lo nd i l l
y R i ve r
Walla
Campbellt
BMarulan
Goulburn
Crookwell
LAKEGEORGE
CANBERRA
Moss Vale
Mittagong
Picton
AvoDa
LAKEBURRAGORANG
WarragambaDam
Katoomba
Blackheath
Wi
Lithgow
Penr
Ko
wm
un
gR
i v
e r
Co
xs
R i ve
rG ro s e
R i ve r
Nap
ea
nR
ive
r
OH
WingeReser
Fitzroy FallsReservoir
LAKE YARRUNGA
Tallowa Dam
Warragambacatchment
NapeanDam
135 km
56 km
12 km
45 km
40 km
59 km
76 MS5.1.1 Perimeter and area 03/05
Part 1 Straight sided figures 77
Exercises – Part 1
Exercises 1.1 to 1.6 Name ___________________________
Teacher ___________________________
Exercise 1.1 – Area and perimeter
1 Match the word ‘perimeter’ or ‘area’ next to each of the
following situations.
a You are painting a wall. _______________________________
b You wrap some string around some nails __________________
c Planting a crop of wheat. ______________________________
d Drawing the outline of a shape. _________________________
e Fencing a property. ___________________________________
78 MS5.1.1 Perimeter and area 03/05
2 Below are two rectangles.
a Check that they both have the same area.
Show your thinking below.
___________________________________________________
___________________________________________________
b Discuss Ivan’s statement below.
Rectangles with the same area alwayshave the same perimeter.
___________________________________________________
___________________________________________________
___________________________________________________
Part 1 Straight sided figures 79
3 a Find the area and perimeter of the following shaded shapes.
i
________________________________________________
________________________________________________
ii
________________________________________________
________________________________________________
iii
________________________________________________
________________________________________________
b What do all these shapes above have in common?
___________________________________________________
c Look at the three shapes above. What is the name of the shape
that has the biggest area?
___________________________________________________
80 MS5.1.1 Perimeter and area 03/05
d Look at the two shapes below.
The first shape does not have the same perimeter as the
rectangle above. However, the second shape does.
Explain why? _______________________________________
___________________________________________________
___________________________________________________
4 a Use the grid paper provided to draw a square that has the same
area as the shaded rectangle below.
rectangle
grid paper
b Name the shape above that has the smallest perimeter.
___________________________________________________
Part 1 Straight sided figures 81
5 (Harder) Use the diagrams below to answer the following questions.
a
The perimeter of this square is 20 m. How long is each side?
___________________________________________________
b7 m
i The perimeter of this rectangle is 20 m. What is its width?
________________________________________________
________________________________________________
ii How many more rectangles have this same perimeter?
________________________________________________
Draw as many as you can on a blank piece of paper and
send it to your teacher.
82 MS5.1.1 Perimeter and area 03/05
Exercise 1.2 – Simple composite figures
1 Use match sticks to make this rectangle.
Each�match�is�one�unit�long
a Write the correct number in the spaces below.
i The perimeter of this rectangle is ____ units long.
ii The area of this rectangle is ____ square units.
b Reduce the area of this rectangle by two square units, without
changing its perimeter. Draw your answer in the space below.
Part 1 Straight sided figures 83
2 a What is the perimeter of the trapezium, below?
___________________________________________________
5 m
6 m
4.1 m
10 m
This trapezium above has been copied, turned around and joined to
the first trapezium to make the parallelogram below.
b Write correct lengths against all four sides of the parallelogram
above. What is the perimeter of this parallelogram?
___________________________________________________
c Explain why the perimeter of the parallelogram is not double the
perimeter of the trapezium.
___________________________________________________
___________________________________________________
84 MS5.1.1 Perimeter and area 03/05
3 An area of land on an inside bend of the Murray River is
shown below.
120 km
130 km
This land contains a rectangular and triangular plot. The total river
frontage is 100 km and both distances along each side of the bend
are the same.
a On the diagram above, write lengths against the remaining
four sides.
b If no fencing is needed either along the river frontage or
between the two plots, how many kilometres of fencing is
required to enclose the remainder of the property?
___________________________________________________
___________________________________________________
4 A large canvas awning for a paved area is made with two pieces of
material.
7 m
20 m
One of the pieces is shaped like a parallelogram and the other is an
equilateral triangle (three sides equal). The awning needs to be
sewn along the outside edge. How many metres of sewing is
there to do?
_______________________________________________________
Part 1 Straight sided figures 85
5 Use 12 matchsticks to make the following shapes. (You are only
allowed to put the matchsticks at 900 to each other. This means thatyou cannot put the matchsticks together like this ⟨ or this <).
a Make the biggest possible area you can, using all
12 matchsticks. Draw your answer below.
b Make the smallest possible area you can using all
12 matchsticks. Draw your answer below.
6 This is the front view of a house.
15 m
3 m
x
The owner has a length of Christmas lights that he uses every year to
highlight the outside of his house. These lights run along the
ground, up the 3 m walls and over the roof with the two ends
meeting. The string of lights are 36.6 m long. How long is x?
_______________________________________________________
_______________________________________________________
86 MS5.1.1 Perimeter and area 03/05
Exercise 1.3 – Triangles and parallelograms
1 There are many lengths given to you in the following diagrams.
7.2 cm
4 cm
5 cm
9 cm
7 mm
24 m
m25 m
m
A B
a Complete the table below by choosing the correct base and
perpendicular height of each of these triangles.
Triangle Base Perpendicular height
A
B
b i In triangle B, which length do you not need to calculate
its area? ________________________________________
ii Calculate the area of triangle B.
________________________________________________
________________________________________________
________________________________________________
c Calculate the area of triangle A.
___________________________________________________
___________________________________________________
___________________________________________________
Part 1 Straight sided figures 87
d i In triangle A, which length do you not need to calculate its
perimeter? _______________________________________
ii Calculate the perimeter of triangle A.
________________________________________________
________________________________________________
________________________________________________
2 a What is the length of the base and the perpendicular height of
the parallelogram below? ______________________________
___________________________________________________
8 m
m
10 m
m
12 mm
b Calculate the area of the parallelogram above.
___________________________________________________
___________________________________________________
c Calculate its perimeter. ________________________________
___________________________________________________
88 MS5.1.1 Perimeter and area 03/05
3 An orange orchard has a river that is 200 m wide flowing through it.
The property has a 1000 m river frontage.
200 m
Ri v
er
600
m
1500 m
1000 m
a Assuming the river is perfectly straight, what is the name of the
shape that the river cuts through the orchard?
___________________________________________________
b Calculate the area of land that the river covers.
___________________________________________________
___________________________________________________
c Calculate the area of the rectangle.
___________________________________________________
___________________________________________________
d Calculate the area of dry land that can be used for
growing oranges.
___________________________________________________
___________________________________________________
e If one orange tree needs 20 m2 of land to grow properly, how
many orange trees need to be planted so that all of the dry land
is used?
___________________________________________________
Part 1 Straight sided figures 89
Exercise 1.4 – Area of rhombus and a kite
1 Answer the following questions about the diagram below.
3 mm
7 mm
4 mm4 mm
5 mm
8 mm
a What shape is it? _____________________________________
b What are the lengths of both diagonals? ___________________
c i Which lengths do you not need to calculate the area of
this figure? ______________________________________
ii Calculate the area of this figure, using the formula A = 12xy ?
________________________________________________
________________________________________________
90 MS5.1.1 Perimeter and area 03/05
d i The figure above contains two isosceles triangles.
Label the base and height of both these triangles on the
diagrams below.
ii Calculate the area of both these triangles.
________________________________________________
________________________________________________
________________________________________________
________________________________________________
________________________________________________
iii Show that the total of these two areas is equal to the area of
the figure at the beginning of this question.
________________________________________________
________________________________________________
Part 1 Straight sided figures 91
2 a The diagonals of a rhombus measure 14 cm and 20 cm.
Which diagram is most correctly drawn?
A 8 cm
9 cm
11 cm
6 cm
B 7 cm
10 cm
10 cm
7 cm
C 10 cm
7 cm
7 cm
10 cm
b Choose the best word from this list to complete the sentence
below: cut, bisect, meet.
The diagonals of a rhombus _________________ each other.
3 a The diagonals of a rhombus measure 12 mm and 7 mm.
Write the correct dimensions on the diagram below.
b Calculate the area of this rhombus by using the formula A = 12xy .
___________________________________________________
___________________________________________________
___________________________________________________
92 MS5.1.1 Perimeter and area 03/05
4 The rhombus below can be cut up into triangles.
9 cm
12 c
m
12 c
m9 cm
a Draw a sketch of one of these triangles in the space below, then
calculate its area.
___________________________________________________
___________________________________________________
___________________________________________________
b Add the areas of the two triangles together to get the total area
of the rhombus.
___________________________________________________
___________________________________________________
c Show that the formula for the area of a rhombus, A = 12xy ,
gives the same area.
___________________________________________________
___________________________________________________
___________________________________________________
Part 1 Straight sided figures 93
Exercise 1.5 – Area of a trapezium
1 a Write the correct lengths on this trapezium, using the
information provided below.
The parallel sides of this trapezium measure 7 cm and 12 cm.
The perpendicular height of the trapezium is 8 cm.
b Complete the following for the information above.
a = __
b = __
h = __
c Calculate the area of this trapezium above, by using the formula,
A =1
2h(a + b) .
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
94 MS5.1.1 Perimeter and area 03/05
2 The regular hexagon below is divided into two trapeziums.
5 cm
13 cm
3 cm
a How long is each side of the hexagon? ____________________
b These two trapeziums are called isosceles trapeziums.
Why do you think they have this name? ___________________
___________________________________________________
c Calculate the area of one of these trapeziums.
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
d Calculate the area of the regular hexagon.
___________________________________________________
___________________________________________________
Part 1 Straight sided figures 95
3 The shaded side of this bus shelter below is to be used for
advertisements.
4 m
2 m
3 m
a Calculate the area of this ‘advertisement’ side of the bus shelter.
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
b If it cost $100 per square metre to advertise, how much does it
cost to cover this whole side with an advertisement?
___________________________________________________
___________________________________________________
96 MS5.1.1 Perimeter and area 03/05
Exercise 1.6 – Area of composite figures.
1 Calculate the area of the shaded part of the shape below by
answering the following questions.
8 m
2 m
1 m
a Complete, to find the area of the triangle.
b =________, h =__________
Area =12bh
___________________________________________________
___________________________________________________
___________________________________________________
b Calculate the area of the square. (Remember, both sides are the
same.) ______________________________________________
___________________________________________________
c Calculate the area of the shaded part by subtracting one area
from the other.
___________________________________________________
___________________________________________________
Part 1 Straight sided figures 97
2 The diagram below is a sewing pattern for an awning that is to go
over a boat.
2.5 m
2.5
m
1.5
m
a This sewing pattern can be divided into a bottom part and a top
part. Name the shape in each part.
___________________________________________________
b What dimensions do you need to calculate the area of the:
i bottom part ______________________________________
ii top part _________________________________________
c Calculate the area of:
i the bottom part ___________________________________
________________________________________________
ii the top part ______________________________________
________________________________________________
98 MS5.1.1 Perimeter and area 03/05
3 The rectangular wall below is to have a mural painted on it.
The shaded section will have a sky blue background.
1 m
1.1 m
0.5
m
2.5 m
2 m
a Divide the shaded area above into two shapes.
(The shapes must be from the list: square, rectangle, triangle,
rhombus, parallelogram, trapezium because these are the area
formulas you know.)
b Calculate the area of both these shapes.
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
c Find the total area of the shaded section.
___________________________________________________
d The wall is 2 metres high by 2.6 metres wide.
Calculate the area of the whole rectangular wall.
___________________________________________________
___________________________________________________
Part 1 Straight sided figures 99
e i The remaining part of the wall is to be painted apple green.
Describe how you would find this area.
________________________________________________
________________________________________________
ii Calculate this area._________________________________
________________________________________________
4 Use the map supplied in the additional resources to answer the
following question.
The area of Warragamba catchment that supplies Sydney’s water
can be approximated into the area of two trapeziums.
The distances supplied to you on the map are actual distances.
Use these two trapeziums to calculate the approximate area of the
Warragamba catchment.
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
100 MS5.1.1 Perimeter and area 03/05
5 a Calculate the area of the side of this step and landing that needs
to be tiled.
90 cm
120 cm15 c
m
15 c
m
Area to be tiled
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
b If tiles cost $42 per square metre, how much will it cost to buy
the tiles?
___________________________________________________
___________________________________________________
Mathematics Stage 5
MS5.1.1 Perimeter and area
Part 2 Figures with curved sides
Part 2 Figures with curved sides 1
Contents – Part 2
Introduction – Part 2..........................................................3
Indicators ...................................................................................3
Preliminary quiz – Part 2 ...................................................5
Circumference and perimeter..........................................13
Areas of circular shapes..................................................17
Quadrants .......................................................................21
Area of a quadrant ..................................................................25
Curved composite figures................................................29
Suggested answers – Part 2 ...........................................35
Exercises – Part 2 ...........................................................47
2 MS5.1.1 Perimeter and area 03/05
Part 2 Figures with curved sides 3
Introduction – Part 2
In this part you will review the formulas for finding the area and
circumference of a circle. You will also learn to choose appropriate
formulas to find the area and perimeter of composite shapes with curved
sides. This enables you to use the area and circumference formulas for a
circle, semicircle and quadrant.
Indicators
By the end of Part 2, you will have been given the opportunity to work
towards aspects of knowledge and skills including:
• calculating the area of simple composite figures consisting of two
shapes, where at least one of them is a circle, semicircle or quadrant
• calculating the perimeter of simple composite figures consisting of
two shapes, where at least one of them is a circle, semicircle
or quadrant.
By the end of Part 2, you will have been given the opportunity to work
mathematically by:
• dissecting composite shapes into simpler shapes
• solving practical problems involving area of simple composite forms
• asking questions which can be solved by using mathematics.
4 MS5.1.1 Perimeter and area 03/05
Part 2 Figures with curved sides 5
Preliminary quiz – Part 2
Before you start this part, use this preliminary quiz to revise some skills
you will need.
Activity – Preliminary quiz
Try these.
1 To what decimal do you think the arrows are pointing?
Write the correct decimal above the arrow on the number
lines below.
a
0 1
b
0.1 0.2
6 MS5.1.1 Perimeter and area 03/05
2 a To what decimals are the following arrows pointing?
2.3 2.4
i ii iii
i ________________________________________________
ii ________________________________________________
iii ________________________________________________
b Which of the decimals, i, ii and iii, on the number line above,
are approximately equal to (closer to):
2.3? ________________________________________________
2.4? ________________________________________________
3 Round off 23.76 to one decimal place. _______________________
4 Approximate 44.44 correct to one decimal place. _______________
5 Approximate 75.398 to two decimal places. ___________________
6 How many right angles are there in one revolution? ____________
7 Rewrite the following fractions as decimals.
a1
2 _________________________________________________
b1
4 _________________________________________________
c3
4 _________________________________________________
Part 2 Figures with curved sides 7
8 Square the following numbers. Check that you are correct by using
the square key x2 on your calculator.
a 11 _________________________________________________
b 12 _________________________________________________
c 15 _________________________________________________
9 a Write the calculator keys that you must press in order to put six
into the memory of your calculator and then to recall it.
___________________________________________________
b Write the calculator keys that you must press in order to put six
into the memory of your calculator, add seven to this memory
and then recall the answer (13).
___________________________________________________
c Write the keys you must press in order to clear the memory of
your calculator.
___________________________________________________
8 MS5.1.1 Perimeter and area 03/05
10 Write the area formula for each of the plane figures below.
Draw a diagram of each shape showing where each pronumeral, used
in the formula, lies on your diagram. The first one has been done
for you.
Square: A = l2
l
l
rectangle: _______________________________________________
triangle: ________________________________________________
parallelogram: ___________________________________________
Part 2 Figures with curved sides 9
rhombus: _______________________________________________
kite: ___________________________________________________
trapezium _______________________________________________
11 Calculate the area of:
a a square that has a side length of 7 cm.
___________________________________________________
___________________________________________________
b a rectangle that has a length of 9 mm and a breadth of 4 mm.
___________________________________________________
___________________________________________________
c a triangle that has a perpendicular height of 15 mm and a base
length of 8 mm.
___________________________________________________
___________________________________________________
___________________________________________________
10 MS5.1.1 Perimeter and area 03/05
12 Calculate the area of the following figures.
a
9 m
6 m
___________________________________________________
___________________________________________________
b 7 m
m
12 mm
___________________________________________________
___________________________________________________
___________________________________________________
c 17 mm
6 m
m
11 mm
___________________________________________________
___________________________________________________
___________________________________________________
Part 2 Figures with curved sides 11
13 Calculate the perimeter of the following figure.
5 cm
3 cm
7 cm
_______________________________________________________
_______________________________________________________
14 Calculate the area and perimeter of the following figure.
6 m2
m
10 m
8 m
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
12 MS5.1.1 Perimeter and area 03/05
Part 2 Figures with curved sides 13
Circumference and perimeter
The circumference of a circle is the length of the outside of a whole
circle. It is a length, just like perimeter. Can you remember how to find
the circumference of a circle?
Circumference = π × diameter
or = 2 × π × radius
circumference
diameter
centre
radius
(Remember, π is pronounced ‘pie’, as in ‘meat pie’, but it is actually
spelt, ‘pi’. π is the letter ‘p’ of the Greek alphabet.)
π is approximately equal to three (π ≈ 3 ), so if the diameter of a circle
is equal to 10 mm, then its circumference is approximately equal to
3×10 = 30 mm.
You can use this quick check method to either check if your calculator
has given a reasonable answer or as an approximation before you do a
more accurate calculation.
14 MS5.1.1 Perimeter and area 03/05
Follow through the steps in this example. Do your own working in the margin
if you wish.
A circle has a radius of 12 mm.
a What is the length of its diameter?
b Using π ≈ 3 , give a rough estimate of the length of the
circumference.
c Using the pi key, , on your calculator, find the
circumference of the circle correct to one decimal place.
d Explain why your more accurate answer above is always
going to be bigger than your rough estimate.
Solution
a The diameter of a circle is twice the radius. So the diameter is
2 × 12 = 24 mm long.
b Using, π ≈ 3 then,
Circumference = π × diameter
or C = πd
≈ 3× 24
C ≈ 72 mm
c C = πd
= π × 24
≈ 75.398 mm
∴ Circumference ≈ 75.4 mm. (Remember, when
approximating this answer you must look at the nine next to the
three. This number is closer to 75.4, not 75.3
d The actual value of π is just a little bigger than three, so π × d
is always going to bigger than triple the diameter.
After calculating the circumference of a circle, you can always do a quick
check by multiplying its diameter by 3.
Use these skills in the following activity to help you calculate the
perimeter of a semicircle.
Part 2 Figures with curved sides 15
Activity – Circumference and perimeter
Try these.
1 A circle has a diameter of 7 mm.
a Use the formula, C = πd and the pi key, to calculate the
length of its circumference.
_______________________________________________________
_______________________________________________________
_______________________________________________________
b What is the approximate length of the circumference if you use
π ≈ 3 ?______________________________________________
Was your first answer close to this approximation?___________
2 This semicircle has the same diameter as the circle above.
7 mm
a i Divide your more accurate value above, by 2 and write your
answer here.______________________________________
ii Explain why this calculation gives the length of the
semicircular arc.
________________________________________________
________________________________________________
________________________________________________
16 MS5.1.1 Perimeter and area 03/05
b i To calculate the perimeter of this semicircle you must add 7
to your answer above. Explain why. __________________
________________________________________________
________________________________________________
ii Complete the equation below.
_____________ = semicircular arc+ diameter
= _____________________
= _____________________
Check your response by going to the suggested answers section.
Always remember to halve the circumference when calculating the length
of the semicircular arc. When calculating the perimeter of a semicircle
you must remember to add the length of the diameter.
You have been revising finding the circumference of circles and the
perimeter of semicircles. Now check that you can solve these kinds of
problems by yourself.
Go to the exercises section and complete Exercise 2.1 – Circumference
and perimeter.
Part 2 Figures with curved sides 17
Areas of circular shapes
Can you remember how to calculate the area of a circle?
Area of a circle = π × radius × radius
= πr2or A
How can you calculate the area of a semicircle?
Or you could halve it.
Isn’t that the same thing.
First you have to find thearea of the whole circleand then divide it by two.
Yeah. You could multiplythe whole circle area byzero point five. That’s thesame thing as well.
radius
Area of semicircle =1
2× area of whole circle
=1
2× π × radius× radius
=1
2πr2 or πr2 ÷ 2 or 0.5πr2
Use the following activity to review your understanding of calculating
areas of circles and semicircles.
18 MS5.1.1 Perimeter and area 03/05
Activity – Areas of circular shapes.
Try these.
1 a Calculate the area of a circle that has a radius of 20 mm.
___________________________________________________
___________________________________________________
___________________________________________________
b Use a pair of compasses to draw a semicircle that has a radius of
20 mm. Sit your semicircle anywhere on the line below.
c Calculate the area of this semicircle.
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
Check your response by going to the suggested answers section.
You must be careful when using your calculator to find areas of circles
and semicircles. Try not to reuse approximated answers.
Put answers that you may need in memory or perhaps do all your
calculations at the end. This will help you get the most accurate answers.
Part 2 Figures with curved sides 19
You have been revising finding areas of circles and semicircles.
Now check that you can solve these kinds of problems by yourself.
Go to the exercises section and complete Exercise 2.2 – Areas of circular
shapes.
20 MS5.1.1 Perimeter and area 03/05
Part 2 Figures with curved sides 21
Quadrants
Remember, a quadrant is just another name for a quarter of a circle.
The teacher draws the following diagram on the board.
centre of circle5 cm
arcquadrant
The shaded part of this diagram is called a quadrant.
The curved part is called an arc.
Explain how you would calculatethe perimeter of this quadrantcorrect to one decimal place.
The little square in the cornermeans it is 90°, so it is a quarterof a circle.
The class checks that 900 × 4 = 3600 .
Ivan then doubles 5 cm to get the diameter and then writes these two
calculations down.
Circumference = π × diameter
= π ×10
Length of arc = π ×10 ÷ 4
22 MS5.1.1 Perimeter and area 03/05
He then uses his calculator like this: 10 ÷ 4 = and gets
7.8539816….
The perimeter is 7.9 cm.
But isn’t the perimeter thetotal of all the sides?
Oh, I forgot. 7.9 + 10 is 17.9.The perimeter is 17.9 cm.
Ivan added 10 to 7.9 because each radius on the quadrant above is 5 cm
long. Ivan’s perimeter calculation is shown below.
Perimeter = arc length + radius+ radius
or P = arc+ 2 × r
= 7.9 + 2 × 5
= 7.9 +10
=17.9 cm
The activity below helps you to remember to add all three sides of the
quadrant together to find its perimeter.
Part 2 Figures with curved sides 23
Activity – Quadrants
Try these.
1 Answer the following questions about the quadrant below.
4.2 mm
a How long is the diameter of the circle that this quadrant came
from?_______________________________________________
b Write the correct numbers in the calculation below.
Circumference = π × diameter
= π × _______
Length of arc = π × ______÷ ________
c Use a calculator to find the length of the arc of this quadrant.
___________________________________________________
d You are asked to find the perimeter of the quadrant above.
Write the missing words and numbers below.
P__________ = _____ length + _________ + radius
= ______+ _______+ _______
e What is the perimeter of the quadrant? (You can use a
calculator or your head.) ________________________________
Check your response by going to the suggested answers section.
In the activity above you calculated the length of the arc of a quadrant by
dividing the circumference of a whole circle by four.
24 MS5.1.1 Perimeter and area 03/05
There are two other ways you could have done this:
• Length of arc = 1
4× π × 8.4
1 ab⁄c 4 8.4 =
or
• Length of arc = 0.25 × π × 8.4
0.25 8.4 =
(Remember, 0.25 is the same as 1
4)
Use one of these methods in the following activity.
Activity – Quadrants
Try these.
2 A colourful awning at the entranceway of a building is supported by
wire that is anchored to the building at both ends.
6 m
awning
wire support
anchor points
How long is the wire, correct to one decimal place?
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
Part 2 Figures with curved sides 25
The question above did not ask for perimeter. Always read the question
carefully, so you do not do extra work.
Area of a quadrant
When calculating the area of a quadrant, you can use similar methods to
the above.
So, for a quadrant, you could firstfind the area of the whole circle andthen divide by four … or multiply byone quarter … or multiply by zeropoint two five.
radius
Area of quadrant = 1
4× area of whole circle
= 1
4× π × radius× radius
= 1
4πr2 or πr2 ÷ 4 or 0.25πr2
The activity below helps you to practise using all three methods so you
can decide on the one you like best.
26 MS5.1.1 Perimeter and area 03/05
Activity – Quadrants
Try these.
3 Answer the following questions about the quadrant below.
82 m
m
a Write the correct numbers and symbols to match the calculator
buttons used to find the area of the quadrant.
i Area of quadrant = ________________________________
1 ab⁄c 4 82 82 =
ii Area of quadrant = ________________________________
0.25 82 x2 =
iii Area of quadrant = ________________________________
82 x2 ÷ 4 =
b Write the answer next to each calculator method above. Do they
all give the same answer?_______________________________
Check your response by going to the suggested answers section.
It doesn’t really matter which method you use as they all give the same
answer. The important thing is to remember to divide the area of the
whole circle by four or to multiply it by 1
4 or 0.25. If you forget to do
this you will get the wrong answer.
Use one of the methods above in this activity.
Part 2 Figures with curved sides 27
Activity – Quadrants
Try this.
4 Calculate the area of the figure below.
3.2 cm
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
Although the formula for the area of a circle is different to the formula
for its circumference, you must remember that for a quadrant, both
formulas must be altered by:
• ÷ 4 or
• × 1
4 or
• × 0.25
You have been calculating the perimeter and area of quadrants.
Now check that you can solve these kinds of problems by yourself.
Go to the exercises section and complete Exercise 2.3 – Quadrants.
28 MS5.1.1 Perimeter and area 03/05
Part 2 Figures with curved sides 29
Curved composite figures
Combining different shapes together in the one diagram produces
composite figures. Look at the diagrams below. Two figures are used
to create each shape. What are they?
In the first diagram you have a semicircle on top of an isosceles
trapezium and in the second you have a square with a quadrant
(1
4 or 0.25 of a circle) removed from it.
You may be asked to find both area and perimeter of these types
of shapes.
The following example shows you how to calculate both the area and
perimeter of the second diagram above.
30 MS5.1.1 Perimeter and area 03/05
Follow through the steps in this example. Do your own working in the
margin if you wish.
Two equal perpendicular sides and an arc enclose the area
below. The radius of the arc is the same length as the two
perpendicular sides.
10 mm
a Calculate the area of this figure correct to one
decimal place.
b Calculate its perimeter correct to one decimal place.
Solution
a The area of this figure looks like it is going to be difficult
to calculate. However, it is only a square with a quadrant
cut out of it. You must subtract the area of the quadrant
from the area of the square. (Complete the 10 mm square
on the diagram above as it will help you visualise the two
figures. Your diagram should look like the previous
diagram.)
Area of square = side× side
=10 ×10
=100 m2
Part 2 Figures with curved sides 31
Area of quadrant = 1
4× area of circle
= 1
4× πr2
=1
4× π ×102 (Note: 102 =10 ×10)
= π ×100 ÷ 4
≈ 78.5 m2
(Remember, you can multiply ten by ten or use the square
key x2 like this: 10 x2 ÷ 4 = )
Area of figure = square area − quadrant area
≈ 100 − 78.5
≈ 21.5 m2
b The perimeter of the figure is easier to visualise.
Perimeter = side + side + length of arc.
First find the diameter because you need it to find the
circumference of the circle. (Diameter = 20 m)
Next calculate the length of the arc.
Length of arc = circumference of circle ÷ 4
= π × diameter ÷ 4
= π × 20 ÷ 4
= π × 5 (because 20 ÷ 4 = 5)
≈ 15.7 m
Finally add the three sides together.
Perimeter = side+ side+ length of arc
=10 +10 +15.7
= 35.7 m
It is important to identify your calculations by using English. Label them
as ‘length of arc =’, or ‘area =’ or ‘perimeter =’. If you don’t, you can
easily make a mistake by adding or subtracting the wrong numbers.
In the next activity, make your mathematics clear by using English to
label your calculations.
32 MS5.1.1 Perimeter and area 03/05
Activity – Curved composite figures
Try these.
1 Answer the following questions about the diagram below.
8 cm
2 cm
4 cm
5 cm
A semicircle and an isosceles trapezium form the diagram above.
a Tick (�) the correct answer. The top section of the diagram is:
a circle with radius 8 cm
a semicircle with radius 8 cm
a circle with radius 4 cm
a semicircle with radius 4 cm.
b Calculate the area of the figure by completing these steps.
i Calculate the area of the top section of the diagram above.
________________________________________________
________________________________________________
________________________________________________
________________________________________________
ii The bottom section of the diagram is a trapezium.
Write down the values of a, b and h, where a and b are the
lengths of the parallel sides and h is the perpendicular
height between the two parallel sides.
a = b = h =
Part 2 Figures with curved sides 33
iii Calculate the area of the trapezium above using the formula
for the area of a trapezium, A =1
2h(a + b) .
________________________________________________
________________________________________________
________________________________________________
iv What is the total area of this composite figure?
________________________________________________
c Calculate the perimeter of the shape by completing these steps.
i Calculate the length of the arc.
________________________________________________
________________________________________________
________________________________________________
ii Calculate the perimeter of the figure.
________________________________________________
________________________________________________
________________________________________________
iii There are many lengths given in the diagram above.
Which lengths did you not need to use in order to calculate
the perimeter of the figure? __________________________
Check your response by going to the suggested answers section.
34 MS5.1.1 Perimeter and area 03/05
When calculating area and perimeter of composite shapes you must:
• recognise the type of shapes you have in the diagram
• recall the formulas needed for calculations
• link the correct dimension with each part of your formula
• decide what to add and what to subtract
• decide when to add and when to subtract.
Practise finding area and perimeter of composite shapes in the
following exercise.
Go to the exercises section and complete Exercise 2.4 – Curved composite
figures
Part 2 Figures with curved sides 35
Suggested answers – Part 2
Check your responses to the preliminary quiz and activities against these
suggested answers. Your answers should be similar. If your answers are
very different or if you do not understand an answer, contact your teacher.
Activity – Preliminary quiz
1 a about 0.7. Halfway between 0 and 1 is 0.5. The arrow is to the
right of 0.5.
b about 0.17. Halfway between 0.1 and 0.2 is 0.15. (You can use
a calculator to check if you wish using (0.1 + 0.2) ÷ 2 = 0.15.)
To the right of this is 0.16, 0.17, 0.18 and 0.19. The next
decimal after this is 0.20 (or simply 0.2).
2 a i 2.33 ii 2.35 iii 2.38
b 2.33 is approximately equal to 2.3 because it is closer to that end
of the number line shown.
It is written like this: 2.33 ≈ 2.3 or 2.33 2.3
2.35 and 2.38 are approximately equal to 2.4 because they are
closer to that end. Although 2.35 is exactly in the middle of 2.3
and 2.4, all mathematicians have agreed that this will always be
approximately equal to the decimal above (in this case 2.4)
rather than the decimal below.
3 23.8
23.7 23.8
23.76
4 44.4
The last four is less than five, so remove the last four and do nothing.
Just leave it alone.
Note: ‘round off’ and ‘approximate’ mean exactly the same thing.
36 MS5.1.1 Perimeter and area 03/05
5 75.40 (Do not remove the zero. This answer is correct to two
decimal places.)
To approximate 75.398 to two decimal places, you must look at the
third decimal place. As eight is bigger than or equal to five, then the
0.39 is changed to 0.40.
Another way of looking at this more difficult number is to imagine
the decimal part of the answer as 398. The closest ten to this number
is 400, not 390.
6 Four
7 a 0.5 Think of 1
2 as 1÷ 2 = 0.5 (Use a calculator to check this.)
Alternatively you can look at the number line at the beginning of the
preliminary quiz. 0.5 is halfway between 0 and 1, just like a half is
also halfway between 0 and 1.
b 0.25 Similarly 1
4 is the same as 1÷ 4 = 0.5
c 0.75 (Same as 3÷ 4 )
8 a 121. When you use a calculator: press 11 x2 =
b 144
c 225
15 ×15 =15 ×10 +15 × 5
=150 + 75
= 225
15 × 5 may also be broken down the same way, like this:
Part 2 Figures with curved sides 37
15 × 5 =10 × 5 + 5 × 5
= 50 + 25
= 75
9 a Some calculators require different key sequences.
Check your calculator manual. One method is shown below.
6 STO M+ C RCL M+
You do not need to press C . This is only included so you can
see the six come and go from your display.
b 6 STO M+ 7 M+ RCL M+
c 0 STO M+
This is how your calculator works:
M+STO C RCL M+6 7 M+ RCL M+ 0 STO M+
replaces memorywith 6
clears the display only
recalls the numberin memory and putsit into the display
adds 7to memory
recalls 6 + 7
replaces memorywith 0
Your calculator may use different keys. If you cannot do this,
check how to use your calculator with your teacher.
When you reuse an answer it is best to put this into your
calculator memory so you don’t lose it.
If you choose to put answers into your calculator memory, you
need to practise these skills.
38 MS5.1.1 Perimeter and area 03/05
10 Rectangle: area = length × breadth
A = lb
l
b
Triangle: area =
1
2× base× height
A =1
2bh
b
h
Parallelogram: area = length × height
A = lhh
l
Rhombus: area =1
2× diagonal× other diagonal
A =1
2xy
x
y
Kite: area =1
2× diagonal× diagonal
A =1
2xy
x
y
Trapezium: area =1
2× height × sum of parallel sides
A =1
2h(a + b)
a
b
h
Part 2 Figures with curved sides 39
11 a 49 cm2
l = 7 Area = l2
= 72
= 7 × 7
= 49 cm2
b 36 mm2
l = 9, b = 4 Area = lb
= 9 × 4
= 36 mm2
c 60 mm2
b = 8, h = 15 Area =1
2bh
=1
2× 8 ×15
= 4 ×15 (by halving 8)
= 60 mm2
12 a 54 m2
l = 6, h = 9 Area = lh
= 6 × 9
= 54 m2
b 42 mm2
x = 12, y = 7 Area =1
2xy
=1
2×12 × 7
= 6 × 7 (by halving 12)
= 42 mm2
40 MS5.1.1 Perimeter and area 03/05
c 84 mm2
h = 6, a = 11, b = 17 Area =1
2h(a + b)
=1
2× 6 × (11+17)
=1
2× 6 × 28
=14 × 6 (by halving 28, the bigger number)
= 84 mm2
13 18 cm. Two sides of this figure are equal. Starting at the top left
corner and going clockwise around the shape,
Perimeter = 3+ 3+ 7 + 5
= 18 cm
14 Area = 56 m2
Perimeter = 36 m
6 m
2 m
10 m
8 m 8 –
2 =
6 m
10 – 6 = 4 m
A
B
You can calculate the area of this shape many ways. Two methods
are explained below.
The first way is to divide the figure into two different areas, A and
B. Calculate the area of each, making sure you work out the
dimensions of each rectangle first.
Area A has dimensions 6 m by 6 m, and area B has dimensions 10 m
by 2 m. (It is not necessary to know the length 10 – 6 = 4 m.)
Total area = area A+ area B
= (6 × 6)+ (10 × 2)
= 36 + 20
= 56 m
Part 2 Figures with curved sides 41
Alternatively you can calculate the area of the big rectangle (area A)
below and take the area of the little rectangle from it.
area A = 80 m2
10 m
8 m
area B = 24 m2
4 m6
m
Area = 80 − 24
= 56 m2
Perimeter = 6 + 6 + 4 + 2 +10 + 8
= 36 m
Make sure you check that you have added six numbers together.
Alternatively you can just double 8 + 10. (This is explored further in
part 1.)
Activity – Circumference and perimeter
1 a about 22 mm
C = πd
= π × 7
You might press these keys: 7 =
Your answer is approximately 21.99 mm.
b 21mm. ( 3× 7 )
If your original answer is not close to 21, you need to recalculate
it. Perhaps you accidentally pressed the wrong keys!
2 a i 10.995. ( 21.99 ÷ 2 ) Your answer could also be 11 mm.
( 22 ÷ 2 ) but because you are reusing answers the more
accurate answer is preferable at this stage.
Note: if you prefer you could use the fraction key on your
calculator like this:
1 ab⁄c 2 21.99 =
42 MS5.1.1 Perimeter and area 03/05
ii A semicircle is half a circle, so the length of the
semicircular arc = 1
2 the whole circumference.
Finding half of something is the same as dividing by two.
b i Perimeter is the total length around the outside of a plane
shape, so you must add the length of the diameter to the
length of the semicircular arc.
ii The first answer below is better because it is always best to
approximate at the end of a calculation than in the middle.
In this question it did not matter but sometimes it does and
then you will get the wrong answer.
Perimeter = semicircular arc+ diameter
=10.995 + 7
=17.995
≈ 18 mm
or
perimeter = semicircular arc+ diameter
=11+ 7
=18 mm
Activity – Areas of circular shapes
1 a 1256.6 mm2 (or perhaps 1257 mm2 )
r = 20 mm, A = ?
Area = πr2
= π × 202
= π × 20 × 20
≈ 1256.6 mm2
You may either simply multiply your pi key, , by 20 then by
20 again or use the square key x2 by pressing 20 x2 =
(Note: it would be a good idea to put this answer into your
calculator memory – see preliminary test answers.)
Part 2 Figures with curved sides 43
b You must open the compass to a radius of 20 mm by pushing the
pencil and the point of the compass against a ruler to help you
measure this radius.
0 10 20 30 40
29 28 27 2630
Put the compass point anywhere in the middle of the line and
draw a big arc that meets the line twice. (See diagram below.)
20 mm
c Area of semicircle = 628.3 mm2 . If you had put the area for the
whole circle, above, into your calculator memory, all you would
need to do is to divide it by two. Otherwise, you may do the
calculation below:
Area of semicircle =1
2πr2
=1
2× π × 202
There are many ways you can use your calculator to get the
above answer. Here is one. 1 ab⁄c 2 20 x2 =
44 MS5.1.1 Perimeter and area 03/05
Activity – Quadrants
1 a 8.4 mm (Double 4.2)
b Circumference = π × diameter
= π × 8.4
Length of arc = π × 8.4 ÷ 4
c about 6.6 mm. The number in your display should be
6.597344573…. To approximate to one decimal place, you
need to look at the nine next to the five. This nine makes the
number closer to 6.6, not 6.5.
d Perimeter = arc length + radius + radius
= 6.6 + 4.2 + 4.2
e 15 mm
2 9.4 m
First double the radius, so
diameter = 2 × 6
=12 mm
Length of arc = π × diameter ÷ 4
= π ×12 ÷ 4
= 9.42477796....
≈ 9.4 m
3 a i1
4× π × 82 × 82 =
ii 0.25 × π × 822 =
iii π × 822 ÷ 4 =
b All answers are the same.Area of quadrant ≈ 5281 square millimetres (mm2 )
4 8.04 cm2 (or 8.0 cm2 )
Area of quadrant = π × radius2 ÷ 4
= π × 3.22 ÷ 4
3.2 x2 ÷ 4 =
Part 2 Figures with curved sides 45
You could also use:
area of quadrant =1
4× π × 3.22
1 ab⁄c 4 3.2 x2 =
or
area of quadrant = 0.25 × π × 3.22
0.25 3.2 x2 =
Activity – Curved composite figures
1 a � a semicircle with radius 4 cm.
b i 25 cm2 (You can choose to write this answer to one
decimal place if you wish.)
Area of semicircle =1
2πr2
=1
2× π × 42
=1
2× π × 4 × 4
= (1
2× 4 × 4)× π
= 8π
≈ 25 cm2
ii a = 2, b = 8, h = 4
iii 20 cm2
Area of trapezium =1
2h(a + b)
=1
2× 4 × (2 + 8)
=1
2× 4 ×10
= 2 ×10 (by halving 4)
= 20 cm2
Alternatively, you can do this in your head.
First add the two parallel sides together (8 + 2 = 10)
46 MS5.1.1 Perimeter and area 03/05
Then halve either 10 or the height, 4. (Halving 10 is better
because it is bigger – It is always easier to work with
smaller numbers in your head.) So 5 × 4 is 20.
iv about 45 cm2
Total area = semicircle+ trapezium
≈ 25 + 20
≈ 45 cm2
c i 12.6 cm
Semicircular arc length =1
2πd (Remember, d=diameter)
=1
2× π × 8
= 4π (by halving 8)
≈ 12.6 cm
(Remember, you could also divide the whole circle
circumference by two.)
ii 24.6 cm
Perimeter = semicircular arc+ three sides of trapezium
≈ 12.6 + 5 + 2 + 5
≈ 24.6 cm
iii 4 cm (height of trapezium) and/or
8 cm (diameter of semicircle).
You need the diameter in order to calculate the length of the
arc, so if you didn’t include 8 cm in your answer, it is still
correct.
Part 2 Figures with curved sides 47
Exercises – Part 2
Exercises 2.1 to 2.5 Name ___________________________
Teacher ___________________________
Exercise 2.1 – Circumference and perimeter
1 A circle has a radius of 22 mm.
a How long is its diameter? _______________________________
b Triple this diameter to get a rough estimate of the length of the
circumference.________________________________________
c Calculate a more accurate value for the circumference of this
circle by using the pi key, , on your calculator.
Write your answer correct to one decimal place.
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
48 MS5.1.1 Perimeter and area 03/05
2 Use this semicircle to complete the task below.
9 cm
a Calculate the length of the semicircular arc.
___________________________________________________
___________________________________________________
___________________________________________________
b Calculate the perimeter of the semicircle.
___________________________________________________
___________________________________________________
3 A circular running track is 119 m wide at its inner most point.
The inner circumference is the smallest distance that you can run if
you complete one lap of this track.
119 m
a The circumference of the inner circle is about 374 m.
Write down the correct working for this answer.
___________________________________________________
___________________________________________________
Part 2 Figures with curved sides 49
b A 1500 m walk is to be organised. Use your answer above to
calculate how many laps of this track competitors must walk.
___________________________________________________
___________________________________________________
c i If the track is 7 m wide, what is the diameter of the outside
edge of the track? (Remember, the track is on both sides if
the inner diameter.) ________________________________
ii Calculate the circumference of the outside edge of the track?
________________________________________________
________________________________________________
________________________________________________
d Why do you think competitors in a long race like to crowd
along the inner edge of running track?
___________________________________________________
___________________________________________________
4 The circumference of a circle is 47 metres long.
Approximately how long is its diameter?
_______________________________________________________
_______________________________________________________
50 MS5.1.1 Perimeter and area 03/05
Exercise 2.2 – Areas of circular shapes
1 a Calculate the area of a circle that has a radius of 12 m.
___________________________________________________
___________________________________________________
___________________________________________________
b Another circle has a diameter of 12 m.
i Tick (�) the answer you think is correct.
The area of this circle is:
the same as the area of the circle above.
half as big as the area of the circle above.
a quarter of the area of the circle above.
ii Check your answer above by calculating the area of this
circle.
________________________________________________
________________________________________________
________________________________________________
________________________________________________
Did you tick the correct statement?____________________
You may change your answer above, if you wish.
Part 2 Figures with curved sides 51
2 The fan below is semicircular. It has a diameter of 0.3 m and it is
made out of a beautiful ivory-coloured lace.
a What is the radius of the fan? ____________________________
b Ignoring the extra material needed to anchor the fan to its arms,
how much material in square metres is needed to make:
i one fan. (Write your answer correct to three
decimal places.)
________________________________________________
________________________________________________
________________________________________________
ii 500 fans. (Write your answer to the nearest square metre.)
________________________________________________
________________________________________________
________________________________________________
c If this special lace costs $150 per square metre, how much
would it cost to buy the material for all 500 fans?
___________________________________________________
___________________________________________________
(Note: this calculation represents the minimum about of material
needed. You would always need to buy more to allow for wastage.)
52 MS5.1.1 Perimeter and area 03/05
3 Young children who do not like getting water in their eyes can use a
special plastic bath hat that protects their face. This hat has a hole in
the middle so that the child’s hair can be washed.
15 cm 15 cm
The diameter of the hole is 15 cm and it protects the eyes with a
15 cm brim.
a i What is the diameter of the hat? (Remember, the hat has
the same brim all the way around it.
________________________________________________
ii What is the radius of the hat? ________________________
iii This hat is first made as a complete circle (without the
hole.) What is the total area of plastic needed to make this
complete circle?
________________________________________________
________________________________________________
________________________________________________
b i What is the radius of the hole where the child’s hair gets
washed?_________________________________________
ii What is the area of this hole?
________________________________________________
________________________________________________
________________________________________________
Part 2 Figures with curved sides 53
c Describe how you would calculate the area of plastic that is in
the finished bath hat?
___________________________________________________
___________________________________________________
54 MS5.1.1 Perimeter and area 03/05
Exercise 2.3 – Quadrants
1 a Use a pair of compasses to draw a quadrant that has a radius of
4 cm. Draw this figure as accurately as you can below.
b Estimate the length of the arc of this quadrant. ______________
c How long is the diameter of the circle that this quadrant
came from?__________________________________________
d i Write the correct numbers and symbols below to show the
method you are going to use.
Circumference of circle = π × diameter
Length of arc = _____________________
ii Follow your method above by pressing the appropriate keys
on your calculator. Write your answer correct to one
decimal place here. ________________________________
e Calculate the perimeter of this quadrant by completing the
following:
perimeter = length of arc + radius + radius
___________________________________________________
___________________________________________________
Part 2 Figures with curved sides 55
2 The dimensions of a colourful awning for an entrance to a building
are shown below.
6 m
awning
Complete the calculation for the amount of material needed to make
this awning.
Area of whole circle = π × radius × radius
Area of quadrant = ________________________________________
_______________________________________________________
_______________________________________________________
3 Answer the following questions about the diagram below.
8 mmx°
a Draw a dotted line on the diagram below to complete the
missing part of the whole circle.
b What is the diameter of this circle? _______________________
c i Calculate the circumference of the whole circle, using the
formula: circumference of whole circle = π × diameter
________________________________________________
________________________________________________
________________________________________________
56 MS5.1.1 Perimeter and area 03/05
d i What fraction of the whole circle is missing in the diagram?
________________________________________________
ii Express this fraction as a decimal.
________________________________________________
iii Calculate the length of the ‘dotted’ arc in the diagram
above, using either the decimal or the fraction above.
________________________________________________
________________________________________________
e Calculate the length of the big arc by completing the method
below. (Choose the correct numbers from above.)
Length of big arc = circumference of circle − length of dotted arc
___________________________________________________
___________________________________________________
f Calculate the perimeter of the figure.
___________________________________________________
___________________________________________________
Part 2 Figures with curved sides 57
Exercise 2.4 – Curved composite figures
1 a Divide this figure into two shapes by drawing one line.
The two shapes must be from this list: triangle, rectangle,
square, rhombus, kite, trapezium, circle, semicircle, quadrant.
AB
C D
b Name the two shapes __________________________________
c Name the dimensions you would need to be given to be able to
find the area of these two shapes. _________________________
___________________________________________________
Draw them on the diagram above.
58 MS5.1.1 Perimeter and area 03/05
2 A circular stained glass window for a church is shown in the
diagram below.
To highlight the cross, the kite surrounding it is made of yellow
stained glass and the rest of the window is blue stained glass.
a If the cross is 1.6 m high and 1 m wide, what is the area of
yellow glass in the kite? (Use the formula, A=1
2xy , where x
and y are the diagonals of the kite.)
___________________________________________________
___________________________________________________
___________________________________________________
b The vertical part of the cross, which is 1.6 m high, divides the
window into two equal sections. What length is the:
i diameter of the circle?______________________________
ii radius of the circle?________________________________
c What is the area of the whole circular window?
Show your working below.
___________________________________________________
___________________________________________________
___________________________________________________
d Calculate the area of blue glass in the window by subtracting the
kite area from the circular area.
___________________________________________________
___________________________________________________
___________________________________________________
Part 2 Figures with curved sides 59
3 A tablecloth is to be made to fit a dining table for a special event.
It will be an enlarged shape of the table because the owner wants it
to hang equally over the edge of the table.
0.6 m1.2 m
0.9 m
a The tablecloth has three shapes inside it. What are the names of
the three shapes? (Two of the shapes have the same name.)
___________________________________________________
b i Two of these shapes can be combined to form one shape.
What is the name of this shape? ______________________
ii Draw a diagram of this shape showing its dimensions.
iii Calculate the area of this shape.
________________________________________________
________________________________________________
________________________________________________
c Calculate the total area of material needed to make this
tablecloth.
___________________________________________________
___________________________________________________
___________________________________________________
60 MS5.1.1 Perimeter and area 03/05
d A lace border is to be sewn around the edge of the table cloth.
Describe in words, the lengths you need to add together, in order
to calculate the amount of lace that will be used.
___________________________________________________
___________________________________________________
e Calculate the amount of lace needed for the tablecloth edging.
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
43675 MS5.1.1 Perimeter and area 61
We need your input! Can you please complete this short evaluation to
provide us with information about this module. This information will
help us to improve the design of these materials for future publications.
1 Did you find the information in the module clear and easy to
understand?
_______________________________________________________
_______________________________________________________
_______________________________________________________
2 What sort of learning activity did you enjoy the most? Why?
_______________________________________________________
_______________________________________________________
3 Name any sections you feel need better explanation (if any).
_______________________________________________________
_______________________________________________________
4 Were you able to complete each part in around 4 hours? If not
which parts took you a longer or shorter time?
_______________________________________________________
_______________________________________________________
5 Do you have access to the appropriate resources? This could include
a computer, graphics calculator, the Internet, equipment and people
to provide information and assist with the learning.
_______________________________________________________
Centre for Learning InnovationNSW Department of Education and Training