Circles. Area Circumference Area of Sectors Perimeter and Area of compound shapes Perimeter and Area...

Circles

Area

Circumference

Area of SectorsPerimeter and

Area of compound shapes

Volume of Spheres and

cones

Radius and Height of Cylinders

Perimeters of sectors

Finding the radius of sectors

Pi Circle words

Volumes of Cylinders

Circle theorems

Rounding Refresher

Area of Segments

Equation of a circle 1

Equation of a circle 2

Simultaneous Equations

Circle formulae

Match the words to the definitions

•Sector

•Segment

•Chord

•Radius

•Arc

•Tangent

•Diameter

•Circumference

•The length around the outside of a circle•A line which just touches a circle at one point•A section of a circle which looks like a slice of pizza•A section circle formed with an arc and a chord•The distance from the centre of a circle to the edge•The distance from one side of a circle to the other (through the centre)•A section of the curved surface of a circle•A straight line connecting two points on the edge of a circle

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Think about circlesThink about a line around the outside of a circleImage that line straightened out- this is the circumference

Pi

People noticed that if you divide the circumference of a circle by the diameter you ALWAYS get the same answer

They called the answer Pi (π) , which is:

3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679 8214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196 4428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273 724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609...

You can use the π button on your

calculator

How many digits can you memorise in 2 minutes?

3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609...

Write down pi!

3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609...

How did you do? What do you think the world record is?

Pi website

http://www.pi-world-ranking-list.com/index.php?page=home

Pi Story • One way to memorise Pi is to write a Pi-em (pi poem)

where the number of letters in each word is the same as the number in pi. For example:

“Now I, even I, would celebrate in rhymes inept,the great immortal Syracusan rivall'd nevermorewho in his wondrous lore passed on beforeleft men his guidance how to circles mensurate.”

Can you write one of your own?

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Rounding to Decimal Places

10 multiple choice questions

0.30.4

30.35

A) B)

C) D)

Round to 1 dp

0.34

0.5

0.49 0.4

0.47

A) B)

C) D)

Round to 1 dp

0.48

2.8 2.74

3.02.7

A) B)

C) D)

Round to 1 dp

2.75

13.4

13.0 14.0

13.3

A) B)

C) D)

Round to 1 dp

13.374

26.5

25.0 26.6

26.0

A) B)

C) D)

Round to 1 dp

26.519

23.1823.20

23.1723.10

A) B)

C) D)

Round to 2 dp

23.1782

500.83

500.80

500.84

500.8A) B)

C) D)

Round to 2 dp

500.8251

0.0040.00417

0.0050.00418

A) B)

C) D)

Round to 3 dp

0.00417

5.00

4.99

4.98

4.90A) B)

C) D)

Round to 2 dp

4.999

0.7300 0.7390

0.73990.7210

A) B)

C) D)

Round to 4 dp

0.72995

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Finding the Circumference

You can find the circumference of a circle by using the formula-

Circumference = π x diameter

For Example-

Area= π x 10 = 31.41592654.... = 31.4 cm (to 1 dp)

10cm

You can find the circumference of a circle by using the formula-

Circumference = π x diameterFor Example-

Area= π x 10 = 31.41592654.... = 31.4 cm (to 1 dp)

10cm

Find the Circumference of a circles with:1. A diameter of :

a) 8cmb) 4cmc) 11cmd) 21cme) 15cm

2. A radius of :a) 6cmb) 32cmc) 18cmd) 24cme) 50cm HOME

1a 25.1cmb 12.6cmc 34.6cmd 66.0cme 47.1cm

2a 37.7cmb 201.1cmc 113.1cmd 150.8cme 157.1cm

ANSWERS

Finding the Area

You can find the area of a circle by using the formula-

Area= π x Radius2

For Example-

Area= π x 72

= π x 49 = 153.93804 = 153.9 (to 1dp) cm2

7cm

Finding the AreaYou can find the area of a circle by using the formula-

Area= π x Radius2

For Example-

Area= π x 72

= π x 49 = 153.93804 = 153.9 (to 1dp) cm2

7cm

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2a 12.6b 78.5c 15.2d 380.1e 314.2f 153.9g 100.5h 28.3

ANSWERS

Finding the Area of a Sector

For Example-The sector here is ¾ of a full circleFind the area of the full circle

Area= π x 72

= π x 49 = 153.93804 = DON’T ROUND YET!

Then find ¾ of that area¾ of 153.93804 = 115.45353 (divide by 4 and multiply by 3)

7cm

To find the area of a sector, you need to work out what fraction of a full circle you have, then work out the area of the full circle and find the fraction of that area.


For Example-The sector here is 3/5 of a full circleFind the area of the full circle

Area= π x 72

= π x 49 = 153.93804 = DON’T ROUND YET!

Then find 3/5 of that area3/5 of 153.93804 = 92.362824 (divide by 5 and multiply by 3)

= 92.4cm2

7cm

Sometimes it is not easy to see what fraction of a full circle you have.You can work it out based on the size of the angle. If a full circle is 360°, and this sector is 216°, the sector is 216/360, which can be simplified to 3/5.

216°

Sometimes the fraction cannot be simplified and will stay over 360

Finding the Area of a SectorThe general formula for finding the area is:

Area of sector= Angle of Sector x πr2

360

Fraction of full circle that sector

covers“of” Area of full

circle

Questions

10cm

260°

11cm

190°

12cm

251°

5cm87°

6.5cm

166°

17cm32°

Find the area of these sectors, to 1 decimal place

1 2 3

654

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1 226.92 200.63 315.44 19.05 61.26 80.7

ANSWERS

Finding the Perimeter of a Sector

For Example-The sector here is ¾ of a full circleFind the area of the full circle

Area= π x 14 (the diameter is twice the radius) = π x 49 = 43.982297...... = DON’T ROUND YET!

Then find ¾ of that circumference¾ of 43.982297...... = 32.99 cm (2 dp)

Remember to add on 7 twice from the straight sides

7cm

To find the perimeter of a sector, you need to work out what fraction of a full circle you have, then work out the circumference of the full circle and find the fraction of that circumference.

You then need to add on the radius twice, as so far you have worked out the length of the curved edge


Sometimes you will not be able to see easily what fraction of the full circle you have.

To find the fraction you put the angle of the sector over 360

Sometimes the fraction cannot be simplified and will stay over 360

250°

This sector is 250/360 or two hundred and fifty, three hundred and sixty-ITHS of the full circle

Simplify if you can

Finding the Perimeter of a SectorThe general formula for finding the area is:

Perimeter of sector= (Angle of Sector x πd) + r + r

360

Fraction of full circle that sector covers “of” Circumference of full

circleDon’t forget the

straight sides

This is the same as d of 2r, but I like r +r as

it helps me remember why we

do it

Questions

10cm

260°

11cm

190°

12cm

251°

5cm87°

6.5cm

166°

17cm32°

Find the perimeter of these sectors, to 1 decimal place

1 2 3

654

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ANSWERS1 65.42 58.53 76.64 17.65 31.86 43.5

Here we will look at shapes made up of triangles, rectangles, semi and quarter circles.

Find the area of the shape below:

10cm

8cm

10cm

Area of this rectangle= 8 x10

=80cm2

Area of this semi circle = π r2 ÷ 2= π x 52 ÷ 2= π x 25 ÷ 2=39.3 cm2 (1dp)

Area of whole shape = 80 + 39.3 = 119.3 cm2

Compound Area and Perimeter


Find the perimeter of the shape below:

10cm

8cm

10cm

Perimeter of this rectangle= 8 + 8 + 10

=26cm(don’t include the red side)

Circumference of this semi circle = πd ÷ 2= π x 10 ÷ 2=15.7 cm (1dp)

Perimeter of whole shape = 26 + 15.7

= 31.7 cm


Find the areaof the shape below:

11cm

10cm

Area of this quarter circle = π r2 ÷ 4= π x 52 ÷ 4= π x 25 ÷ 4=19.7 cm2 (1dp)

Area of whole shape = 110+ 19.7 = 129.7cm2

5cm

Area of this rectangle 10 x 11=110


Find the perimeter of the shape below:

11cm

10cm

Work out all missing sides first

Circumference of this quarter circle = πd ÷ 4= π x 10 ÷ 4 (if radius is 5, diameter is 10)=7.9 cm (1dp)

Area of whole shape = 42+ 7.9 = 49.9cm

5cm

6cm5cm

10cm

?

Add all the straight sides=10+10 + 11+ 5 + 6= 42cm

Questions

10cm

11cm

12cm

6cm

20cm

10cm

Find the perimeter and area of these shapes, to 1 decimal place

1 2 3

654

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4cm

17cm

20cm

2cm

6cm

4cm

5cm

12cm

10cm

5cm5cm

Do not worry about perimeter here

Do not worry about perimeter here

ANSWERS AREA PERIMETER

1 38.1 23.42 135.0 61.33 181.1 60.84 27.3

5 129.3 47.76 128.5

Volume of Cylinders

Here we will find the volume of cylinders

Cylinders are prisms with a circular cross sections, there are two steps to find the volume

1) Find the area of the circle

1) Multiple the area of the circle by the height or length of the cylinder

Volume of Cylinders 2

1) Find the area of the circleπ x r2

π x 42 π x 16 = 50.3 cm2 (1dp)


50.3 (use unrounded answer from calculator) x 10 = 503cm3

EXAMPLE- find the volume of this cylinder

10cm

4cm

QuestionsFind the volume of these cylinders, to 1 decimal place

1 2 3

654

4cm

12cm

3cm

10cm

5cm

15cm

3cm

18cm

7cm

14cm

2cm

11.3cm

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1 603.2

2 282.7

3 1178.1

ANSWERS

4 142.0

5 2155.1

6 508.9

Volume of Cylinders 2


π x 42 π x 16 = 50.3 cm2 (1dp)


50.3 x h = 140cm3

Rearrange this to giveh= 140 ÷ 50.3h=2.8 cm

EXAMPLE- find the height of this cylinder

Volume= 140cm3

4cm

h

Volume of Cylinders



π x r2 x 30 = 250cm3

94.2... x r2 = 250Rearrange this to giver2 = 250 ÷ 94.2r2 =2.7 (1dp)r= 1.6 (1dp) cm

EXAMPLE- find the radius of this cylinder

Volume= 250cm3

r

30cm

QuestionsFind the volume of these cylinders, to 1 decimal place

1 2 3

654

4cm

h

3cm

h

5cm

h

r

8cm

r

14cm

r

12cm

volume= 100cm3volume= 120cm3volume= 320cm3

volume= 200cm3 volume= 150cm3volume= 90cm3

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ANSWERS1 6.42 4.23 1.34 2.35 1.86 1.9

Volume of Spheres

The formula for the volume of a sphere is

10cme.g

A= 4/3 x π x 103

A= 4/3 x π x 1000A=4188.8 cm3 (1 dp)

Volume of Cones

The formula for the volume of a cone is

10cme.g

A= 1/3 x π x 42 x 10A= 1/3 x π x 16 x 10A=167.6 cm3 (1 dp)

4cm

10cm

QuestionsFind the volume of these spheres, to 1 decimal place

1 2 3

654

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20cm 5cm

12cm

4cm

13cm

3cm

15cm

9cm

1 4188.82 33510.33 523.6

ANSWERS

4 201.15 122.56 1272.3

Circles Theorems

Angle at the centre

Angles connected by a

chord

Triangles made with a diameter

or radiiCyclic

Quadrilaterals

Tangents

50°

x

Example

Double AngleThe angle at the centre of a circle is twice the angle at the edge

Angle x = 50 x 2 x=100°

25°

x

160°100°

60°

135°

90°

xx

xxx

12 3

64 5

HOME

Answers1) 502)1203)1804)505)67.56)80

90°

Triangles inside circlesA triangle containing a diameter, will be a right angled triangle

A triangle containing two radii, will be isosceles

x

x

60°

x

1 2 3

31 2

72°

x

x x

x y

y

x

100°

x30°

22°

y

Answers1) X=302)x=183)x=454)X=40 y=405)x=30 y= 1206)x=22 y=136

x

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Angles connected by a chord

Angles connected by a chord are equal

x

x

y

y

25°x

12

3

645

y15°

yz

z

x

y

x

z

x

y

y

z

x

25° 53°30°

z

y

x

80°17°

95°35°

40°

125°

15°

40°

10°

100°

Answers1) x=25 y=152)x=125 y= 40 z=153)x=10 y=70 z=1004)X=105 y=40 z=355)x=53 y= 30 z=726)x=85 y=80 z=17

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90°

Tangents to a circleA tangent will always meet a radius at 90°

40°xy

z

3

120°

x

4

140°

x

2

x35 °

1

y

z

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Cyclic QuadrilateralsOpposite angles in a cyclic quadrilaterals add up to 180°

x

y

100° 60°

100 + y = 180 y=80°

60 + x = 180 x = 120 °

x

yx

y

x

y

95° 110° 54°

75°

20°

80°

x 2a

4b

15° 70° a

b

1

25°

yz

w

2 3

4 5Answers1) x=70 y=852)x=126 y=1053)x=100 y=1604)w=15 x=70 y=65 z= 255)a=60 b=36

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Here we will look at finding the area of sectors

You will need to be able to do two things:

Area of Segments

1) Find the area of a sector using the formula-

2) Find the area of a triangle using the formula-

Area= ½ absinC

Area of sector= Angle of Sector x πr2

360

C

b

a

Example-find the area of the blue segment

10cm10cm100°

Step 1- find the area of the whole sectorArea= 100/360 x π x r2

= 100/360 x π x 102

=100/360 x π x 100 =87.3cm2

Step 2- find the area of the triangleArea= ½ absinC =1/2 x 10 x 10 x sin100 = 49.2cm2

Step 3- take the area of the triangle from the area of the segment

87.3 – 49.3 = 38 cm2

Example-find the area of the blue segment

12cm12cm120°

Step 1- find the area of the whole sectorArea= 120/360 x π x r2

= 120/360 x π x 122

=120/360 x π x 144 =150.8cm2

Step 2- find the area of the triangleArea= ½ absinC =1/2 x 12 x 12 x sin120 = 62.4cm2

Step 3- take the area of the triangle from the area of the segment

150.8 – 62.4 = 88.4 cm2

Questions

10cm

130°11cm85°

12cm170°

5cm95°6.5cm

Find the area of the blue segments, to 1 decimal place

1 2 3

654

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17cm65°160°

ANSWERS1 75.12 29.53 201.14 8.35 51.86 33.0

Finding the Radius or angle of a Sector

r

Area= 100 x π x r2

360200= 100 x π x r2

360

200x360 = r2

100 x π

229.2=r2

15.1cm =r

Area=150

10cm x100° Area=200cm2

Area= θ x π x r2

360150= θ x π x 102

360

150x360 = θ102 x π

117.9°= θ

Questions

r

200°

r

175°

r

250°

5cmθ

6.5cm

θ17cm

θ

Find the missing radii and angles of these sectors, to 1 decimal place

1 2 3

654

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Area=100cm2 Area=120cm2Area=50cm2

Area=35cm2

Area=45cm2

Area=120cm2

ANSWERS1 7.62 8.33 5.44 160.45 122.16 47.6

The Equation of a CircleThe general equation for a circle is (x-a)2 + (y-b)2=r2

This equation will give a circle whose centre is at (a,b) and has a radius of r

For example a circle has the equation (x-2)2 + (y-3)2=52

This equation will give a circle whose centre is at (2,3) and has a radius of 5

The Equation of a CircleA circle has the equation (x-5)2 + (y-7)2=16

This equation will give a circle whose centre is at (5,7) and has a radius of 4 (square root of 16 is 4)

For example a circle has the equation (x+2)2 + (y-4)2=100

This equation will give a circle whose centre is at (-2,4) and has a radius of 10

You could think of this as (x - -2)2

The Equation of a CircleA circle has the equation (x-5)2 + (y-7)2=16

What is y when x is 1?

(1-5)2 + (y-7)2=1612+ (y-7)2=161+ (y-7)2=16(y-7)2=15y-7= ±3.9 (square root of 15 to 1 dp)y= 7±3.9y= 10.9 or 3.1There are two coordinates on the circle with x=1, one is (1,10.9) and the other is (1,3.1)

The Equation of a Circle1) Write down the coordinates of the centre point and radius of each of these circles:

a) (x-5)2 + (y-7)2=16

b) (x-3)2 + (y-8)2=36

c) (x+2)2 + (y-5)2=100

d) (x+2)2 + (y+5)2=49

e) (x-6)2 + (y+4)2=144

f) x2 + y2=4

g) x2 + (y+4)2=121

h) (x-1)2 + (y+14)2 -16=0

i) (x-5)2 + (y-9)2 -10=15

2) What is the diameter of a circle with the equation (x-1)2 + (y+3)2 =64

3) Calculate the area and circumference of the circle with the equation (x-5)2 + (y-7)2=16

4) Calculate the area and perimeter of the circle with the equation (x-3)2 + (y-5)2=16

5) Compare your answers to question 3 and 4, what do you notice, can you explain this?

6 ) A circle has the equation (x+2)2 + (y-4)2=100, find:

a) x when y=7

b) y when x=6HOME

Answers1a) r=4 centre (5,7)b) r=6 centre (3,8)c) r=4 centre (-2,5)d) r=10 centre (-2,-5)e) r=7 centre (6,-4)f) r=12 centre (0,0)g) r=411centre (0,-4)h) r=4 centre (1,-14)i) r=5 centre (5,9)

6a) x= 11.5 or -7.5b) y=11.3 or -3.3

Answers2) 163)Circumference = 25.1 Area=50.34)Circumference = 25.1 Area=50.35) Circles have the same radius but different centres, they are translations

The Equation of a Circle 2

Remember- The general equation for a circle is (x-a)2 + (y-b)2=r2

The skill you will need is called completing the square, you may have used it to solve quadratic equations

Here we will look at rearranging equations to find properties of the circle they represent

The Equation of a Circle 2Example x2 + y2 -6x – 8y =0

Create two brackets and put x in one and y in the other

(x ) 2 + (y ) 2 = 0

Half the coefficients of x and y and put them into the brackets, and then subtract those numbers squared

(x -3) 2 + (y - 4) 2 – 32 - 42= 0Tidy this up(x -3) 2 + (y - 4) 2 – 25= 0(x -3) 2 + (y - 4) 2 = 25

This circle has a radius of 5 and centre of (3,4)

The Equation of a Circle 2Example x2 + y2 -10x – 4y- 7 =0

Create two brackets and put x in one and y in the other

(x ) 2 + (y ) 2 = 0

Half the coefficients of x and y and put them into the brackets, and then subtract those numbers squared

(x -5) 2 + (y - 2) 2 – 52 – 22 - 7= 0Tidy this up(x -5) 2 + (y - 2) 2 – 36= 0(x -5) 2 + (y - 2) 2 = 36

This circle has a radius of 6 and centre of (5,2)

The Equation of a Circle 2

You must always make sure the coefficient of x2 and y2 is 1

You may have to divide through 2x2 + 2y2 -20x – 8y- 14 =0

Divide by 2 to give x2 + y2 -10x – 4y- 7 =0

Then put into the form x2 + y2 -10x – 4y- 7 =0

QuestionsPut this equations into the form (x-a)2 + (y-b)2=r2 then find the centre and radius of the circle

1. x2 + y2 -8x – 4y- 5 =02. x2 + y2 -12x – 6y- 4 =0 3. x2 + y2 -4x – 10y- 20 =0 4. x2 + y2 -10x – 14y- 7 =0 5. x2 + y2 -12x – 2y- 62 =06. 2x2 +2y2 -20x – 20y- 28 =0 7. 3x2 + 3y2 -42x – 24y- 36 =0 8. 5x2 + 5y2 -100x – 30y- 60 =0

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Answers1) r=5 centre (4,2)2) r=7 centre (6,3)3) r=7 centre (2,5)4) r=9 centre (5,7)Answers5) r=10 centre (6,1)6) r=8 centre (5,5)7) r=8 centre (6,4)8) r=11 centre (10,3)


A circle has the equation (x-5)2 + (y-7)2=16 and a line has an equation of y=2x+1, at what points does the line intercept the circle?We need to substitute into the equation of the circle so that we only have x’s or y’s

Because y=2x +1 we can rewrite the equation of the circle but instead of putting “y” in we’ll write “2x+1”

So, (x-5)2 + (2x-1-7)2=16(x-5)2 + (2x-8)2=16 expand the bracketsx2-10x + 25 + 4x2 – 32x +64 = 16 simplify and make one side 05x2 -42x + 73=0 solve this quadratic equation to find x, Put the value / values of x into y=2x+1 to find the coordinates of the intercept / intercepts to answer the question

Ways to solve quadratic equations-

Completing the squareFactorising

The Quadratic formula


A quadratic equation can give 1,2 or no solutions, a line can cross a circle at 1,2 or no points

1 solution to the quadratic-The line is a tangent

0 solutions to the quadratic the circle and the line never meet

2 solutions to the quadratic

Intercepts between lines and circles

1) Find out whether these circles and lines intercept, if they do find the coordinates

of the interceptions

a) (x-5)2 + (y-7)2=16 and y=3x-1

b) (x-3)2 + (y-8)2=36 and y=2x-2

c) (x+2)2 + (y-5)2=100 and y=3x + 3

d) (x+2)2 + (y+5)2=49 and 2y+4=x

e) (x-6)2 + (y+4)2=144 and y -3x =5

ANSWERS (all have been rounded)

(3.6,9.8) and (2.2,5.6)(7.2,12.3) and (2,2)(3.5,13.4) and (-2.7,-5)(-0.4,3.3) and (-6.4,-8.9)(-4.6,6.9) and (-4.6,-8.9)

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Circle Formulae

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Circles. Area Circumference Area of Sectors Perimeter and Area of compound shapes Perimeter and Area...

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Transcript of Circles. Area Circumference Area of Sectors Perimeter and Area of compound shapes Perimeter and Area...