© Boardworks Ltd 2008 1 of 84 Maths Age 11-14 S8 Perimeter, area and volume.

© Boardworks Ltd 2008 1 of 84

Maths Age 11-14

S8 Perimeter, area and volume


A

A

Contents


S8.1 Perimeter

S8.2 Area


Put these shapes in order


Perimeter

To find the perimeter of a shape we add together the length of all the sides.

What is the perimeter of this shape?

1 cm

3

3

2

11

2

Perimeter = 3 + 3 + 2 + 1 + 1 + 2

= 12 cm

Starting point


Perimeter of a rectangle

To calculate the perimeter of a rectangle we can use a formula.

length, l

width, w

Using l for length and w for width,

Perimeter of a rectangle = l + w + l + w

= 2l + 2wor

= 2(l + w)


Perimeter


b cm

a cm

9 cm

5 cm

12 cm 4 cm

The lengths of two of the sides are not given so we have to work them out before we can find the perimeter.

Let’s call the lengths a and b.

Sometimes we are not given the lengths of all the sides. We have to work them out using the information we are given.


Perimeter

9 cm

5 cm

b cm

12 cm

a cm

4 cm

a = 12 – 5 cm

= 7 cm

7 cm

b = 9 – 4 cm

= 5 cm

5 cm

P = 9 + 5 + 4 + 7 + 5 + 12

= 42 cm

Sometime we are not given the lengths of all the sides. We have to work them out from the information we are given.


Calculate the lengths of the missing sides to find the perimeter.

P = 5 + 2 + 1.5 + 6 + 4 + 2 + 10 + 2 + 4 + 6 + 1.5 + 2

Perimeter

5 cm

2 cm

6 cm

4 cm4 cm

2 cm2 cm

p

q r

s

t

u

p = 2 cm

q = r = 1.5 cm

s = 6 cm

t = 2 cm

u =10 cm

= 46 cm


P = 5 + 4 + 4 + 5 + 4 + 4

Perimeter


Remember, the dashes indicate the sides that are the same length.

5 cm

4 cm

= 26 cm


Perimeter

Perimeter = 4.5 + 2 + 1 + 2 + 1 + 2 + 4.5

Start by finding the lengths of all the sides.

5 m

2 m

2 m

2 m

4 m

4.5 m 4.5 m

1 m1 m

= 17 m



Before we can find the perimeter we must convert all the lengths to the same units.

Perimeter

3 m

2.4 m

1.9 m

256 cm

In this example, we can either use metres or centimetres.

Using centimetres,

300 cm

240 cm

190 cm

P = 256 + 190 + 240 + 300

= 986 cm



Equal perimeters

Which shape has a different perimeter from the first shape?

A B C

A B C

A B C

B

A

A


Contents


A

A

S8.2 Area

S8.1 Perimeter


The area of a shape is a measure of how much surface the shape takes up.

Area

For example, which of these rugs covers a larger surface?

Rug A

Rug B

Rug C


Area of a rectangle

Area is measured in square units.

We can use mm2, cm2, m2 or km2.

The 2 tells us that there are two dimensions, length and width.

We can find the area of a rectangle by multiplying the length and the width of the rectangle together.

length, l

width, w

Area of a rectangle

= length × width

= lw


Area of a rectangle

What is the area of this rectangle?

8 cm

4 cm

Area of a rectangle = lw

= 8 cm × 4 cm

= 32 cm2


Area of a right-angled triangle

What proportion of this rectangle has been shaded?

8 cm

4 cm

What is the shape of the shaded part?

What is the area of this right-angled triangle?

Area of the triangle = × 8 × 4 = 12

4 × 4 = 16 cm2


We can use a formula to find the area of a right-angled triangle:


base, b

height, h

Area of a triangle = 12

× base × height

= 12

bh



Calculate the area of this right-angled triangle.

6 cm

8 cm

10 cm

To work out the area of this triangle we only need the length of the base and the height.

We can ignore the third length opposite the right angle.

Area = 12

× base × height

= × 8 × 612

= 24 cm2


Area of shapes made from rectangles

How can we find the area of this shape?

7 m

10 m

8 m

5 m

15 m

15 m

We can think of this shape as being made up of two rectangles.

Either like this …

… or like this.

Label the rectangles A and B.

A

B Area A = 10 × 7 = 70 m2

Area B = 5 × 15 = 75 m2

Total area = 70 + 75 = 145 m2


Area of shapes made from rectangles

How can we find the area of the shaded shape?

We can think of this shape as being made up of one rectangle with another rectangle cut out of it.

7 cm

8 cm

3 cm

4 cm Label the rectangles A and B.

A

B

Area A = 7 × 8 = 56 cm2

Area B = 3 × 4 = 12 cm2

Total area = 56 – 12 = 44 cm2


ED

C

B

A

Area of an irregular shapes on a pegboard

We can divide the shape into right-angled triangles and a square.

Area A = ½ × 2 × 3 = 3 units2

Area B = ½ × 2 × 4 = 4 units2

Area C = ½ × 1 × 3 = 1.5 units2

Area D = ½ × 1 × 2 = 1 unit2

Area E = 1 unit2

Total shaded area = 10.5 units2

How can we find the area of this irregular quadrilateral constructed on a pegboard?


C D

BA


An alternative method would be to construct a rectangle that passes through each of the vertices.

The area of this rectangle is 4 × 5 = 20 units2

The area of the irregular quadrilateral is found by subtracting the area of each of these triangles.




Area A = ½ × 2 × 3 = 3 units2

A B

C D

Area B = ½ × 2 × 4 = 4 units2

Area C = ½ × 1 × 2 = 1 units2

Area D = ½ × 1 × 3 = 1.5 units2

Total shaded area = 9.5 units2

Area of irregular quadrilateral= (20 – 9.5) units2

= 10.5 units2



Area of an irregular shape on a pegboard


Area of a triangle

What proportion of this rectangle has been shaded?

8 cm

4 cm

Drawing a line here might help.

What is the area of this triangle?

Area of the triangle = × 8 × 4 = 12

4 × 4 = 16 cm2


Area of a triangle


Area of a triangle

The area of any triangle can be found using the formula:

Area of a triangle = × base × perpendicular height12

base

perpendicular height

Or using letter symbols:

Area of a triangle = bh12


Area of a triangle

What is the area of this triangle?

Area of a triangle = bh12

7 cm

6 cm

= 12

× 7 × 6

= 21 cm2


Area of a parallelogram



Area of a parallelogram = base × perpendicular height

base


The area of any parallelogram can be found using the formula:


Area of a parallelogram = bh



What is the area of this parallelogram?


12 cm

7 cm

= 7 × 12

= 84 cm2

8 cm

We can ignore this length


Area of a trapezium


Area of a trapezium

The area of any trapezium can be found using the formula:

Area of a trapezium = (sum of parallel sides) × height12


Area of a trapezium = (a + b)h12


a

b


Area of a trapezium

9 m

6 m

14 m


= (6 + 14) × 912

= × 20 × 912

= 90 m2

What is the area of this trapezium?


Area of a trapezium

What is the area of this trapezium?


= (8 + 3) × 1212

= × 11 × 1212

= 66 m2

8 m

3 m

12 m


Area problems

7 cm

10 cm

What is the area of the yellow square?

We can work this out by subtracting the area of the four blue triangles from the area of the whole blue square.

If the height of each blue triangle is 7 cm, then the base is 3 cm.

Area of each blue triangle = ½ × 7 × 3

= ½ × 21

= 10.5 cm2

3 cm

This diagram shows a yellow square inside a blue square.


Area problems

7 cm

10 cm

We can work this out by subtracting the area of the four blue triangles from the area of the whole blue square.

There are four blue triangles so:

Area of four triangles = 4 × 10.5 = 42 cm2

Area of blue square = 10 × 10 = 100 cm2

Area of yellow square = 100 – 42 = 58 cm2

3 cm

This diagram shows a yellow square inside a blue square.

What is the area of the yellow square?


Area formulae of 2-D shapes

You should know the following formulae:

b

hArea of a triangle = bh

12



b

h

a

h

b


Using units in formulae

Remember, when using formulae we must make sure that all values are written in the same units.

For example, find the area of this trapezium.

76 cm

1.24 m

518 mm

Let’s write all the lengths in cm.

518 mm = 51.8 cm

1.24 m = 124 cm

Area of the trapezium = ½(76 + 124) × 51.8

= ½ × 200 × 51.8

= 5180 cm2

Don’t forget to put the units at the end.

© Boardworks Ltd 2008 1 of 84 Maths Age 11-14 S8 Perimeter, area and volume.

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Transcript of © Boardworks Ltd 2008 1 of 84 Maths Age 11-14 S8 Perimeter, area and volume.