404

22C H A P T E R

Circumference andarea of a circle

22.1 Circumference of a circleThe circumference is the special name of the perimeter of a circle, that is, the distance allaround it.

Measure the circumference and diameter of some circular objects.

For each one, work out the value of �circ

d

u

ia

m

m

fe

e

r

t

e

e

n

r

ce�.

The answer is always just over 3

The value of �circ

d

u

ia

m

m

fe

e

r

t

e

e

n

r

ce� is the same for every circle, 3.142

correct to 3 decimal places.

The actual value cannot be found exactly and the Greek letter � (pi) is used to represent it.

So, for all circles,

and

Using C to stand for the circumference of a circle with diameter d,

and

Work out the circumference of a circle with a diameter of 6.8 cm.Give your answer correct to 1 decimal place.

Solution 1

� � 6.8

� 21.3628…

Circumference � 21.4 cm Round the circumference to 1 decimal place. The units (cm) are thesame as the diameter’s.

Write down at least four figures of the calculator display.

Multiply the diameter by �.Use your calculator’s � button, if it has one. Otherwise use 3.142

Example 1

To find the circumference of a circle, multiply its diameter by �.

C � �d�Cd

� � �

circumference � � � diameter�circ

d

u

ia

m

m

f

e

e

t

re

e

n

r

ce�� �

C i r c u m f e r

e

n

c

e

diameter

405

22.1 Circumference of a circle CHAPTER 22

Sometimes the radius, not the diameter, is given in a question. In that case, one way offinding the circumference is to double the radius to obtain the diameter and then multiplythe diameter by �.

Alternatively, use the fact that a circle’s diameter d is twice its radius r, that is, d � 2r.Replace d by 2r in the formula C � �d giving C � � � 2r which can be written as

Work out the circumference of a circle with a radius of 8.2 m.Give your answer correct to 1 decimal place.

Solution 2Method 18.2 � 2 � 16.4

� � 16.4

� 51.522…

Circumference � 51.5 m

Method 22 � � � 8.2

� 51.522…

Circumference � 51.5 m

Sometimes the circumference is given and the diameter or radius has to be found.

The circumference of a circle is 29.4 cm.Work out its diameter. Give your answer correct to 2 decimal places.

Solution 3Method 1

29.4 � �d�d � � � 29.4 ÷ �

d � 29.4 � �� 9.3583…

Diameter � 9.36 cm

The formula C � �d can be rearranged with d as the subject and used to find thediameter of a circle, if its circumference is given.

Dividing both sides of C � �d by � gives

To find the diameter of a circle, divide its circumference by �.

d � ��

C�

Example 3

Example 2

C � 2�r

Double the radius to find the diameter.

Multiply the diameter by �.

Write down at least four figures of the calculator display.

Round the circumference to 1 decimal place. The units are m.

Substitute 8.2 for r in the formula C � 2�r.

Write down at least four figures of the calculator display.

Round the circumference to 1 decimal place. The units are m.

Substitute 29.4 for C in the formula C � �d.Divide both sides by �.Divide 29.4 by � and write down at least four figures of thecalculator display.Round the diameter to 2 decimal places. The units are cm.

406

CHAPTER 22 Circumference and area of a circle

Method 229.4 ÷ �� 9.3583…Diameter � 9.36 cm

Exercise 22A

If your calculator does not have a � button, take the value of � to be 3.142 Give answers correct to 1 decimal place unless stated otherwise.

1 Work out the circumferences of circles with these diameters.

a 4.2 cm b 9.7 m c 29 cm d 12.7 cm e 17 m

2 Work out the circumferences of circles with these radii. Give your answers correct to2 decimal places.

a 3.9 cm b 13 cm c 6.3 m d 29 m e 19.4 cm

3 Work out the diameters of circles with these circumferences.

a 17 cm b 25 m c 23.8 cm d 32.1 cm e 76.3 m

4 The circumference of a circle is 28.7 cm. Work out its radius. Give your answer correctto 2 decimal places.

5 The diameter of the London Eye is 135 m. Work out its circumference. Give youranswer to the nearest metre.

6 The tree with the greatest circumference in the world is a Montezuma cypress tree inMexico. Its circumference is 35.8 m. Work out its diameter.

7 Taking the Equator as a circle of radius 6370 km, work out the length of the Equator.Give your answer correct to 1 significant figure.

8 The circumference of a football is 70 cm. Work out its radius.

9 A semicircle has a diameter of 25 cm.Work out its perimeter.(Hint: the perimeter includes the diameter.)

10 A semicircle has a radius of 19 m.Work out its perimeter.

11 The diagram shows a running track.The ends are semicircles of diameter 57.3 m and the straights are 110 m long.Work out the total perimeter of the track.Give your answer correct to the nearest metre.

12 A reel of cotton has a radius of 1.3 cm. The cotton is wrapped round it 500 times.Work out the total length of cotton. Give your answer in metres.

Divide the circumference by �.Write down at least four figures of the calculator display.Round the diameter to 2 decimal places. The units are cm.

�

25 cm

19 m

57.3 m

110 m

13 The radius of a cylindrical tin of soup is 3.8 cm.Work out the length of the label. (Ignore the overlap.)

14 The diameter of a car wheel is 52 cm.a Work out the circumference of the wheel. Give your answer correct to the nearest

centimetre.b Work out the distance the car travels when the wheel makes 400 complete turns.

Give your answer in metres.

15 The big wheel of a ‘penny-farthing’ bicycle has a radius of 0.75 m.Work out the number of complete turns the big wheel makes when the bicycle travels 1 kilometre.

22.2 Area of a circleThe diagram shows a circle which has been split up into equal sectors.

The sectors can be rearranged to make this new shape.

Splitting the circle up into more and more sectors and rearranging them, the new shape becomes very nearly a rectangle.The length of the rectangle is half the circumference of the circle, and the width of the rectangle is equal to the radius of the circle.

The area of the rectangle is equal to the area of the circle.

Area of circle � �12� circumference � radius

Using A to stand for the area of a circle with radius r,

A � �12� � 2�r � r

This means that the area of a circle is � � radius � radius.

To find the area of a circle, multiply � by the square of the radius.

A � �r2

407

22.2 Area of a circle CHAPTER 22

radius

circumference12

The radius of a circle is 6.7 cm.Work out its area.Give your answer correct to the nearest whole number.

Solution 4� � 6.72

� 141.026…

Area � 141 cm2

If the diameter, not the radius, is given, the first step is to halve the diameter to get theradius.

The diameter of a circle is 9.6 m.Work out its area.Give your answer correct to 1 decimal place.

Solution 59.6 � 2 � 4.8

� � 4.82

� 72.3822…

Area � 72.4 m2

Exercise 22B

If your calculator does not have a � button, take the value of � to be 3.142. Giveanswers correct to 3 significant figures.

1 Work out the areas of circles with these radii.

a 7.2 cm b 14 m c 1.5 cm d 3.7 m e 2.43 cm

2 Work out the areas of circles with these diameters.

a 3.8 cm b 5.9 cm c 18 m d 0.47 m e 7.42 cm

3 The radius of a dartboard is 22.86 cm. Work out its area to the nearest cm2.

4 The diameter of Avebury stone circle is 365 m.Work out the area enclosed by the circle.Give your answer correct to 1 significant figure.

Example 5

Example 4

408

CHAPTER 22 Circumference and area of a circle

Press the calculator keys exactly as shown here and then press

or press 6.7 6.7 and then press .

�

� � ��

Write down at least four figures of the calculator display.

Round the area to the nearest whole number. The units are cm2.

Divide the diameter by 2 to get the radius.

Square the radius and then multiply by �.

Write down at least four figures of the calculator display.

Round the area to 1 decimal place. The units are m2.

�

5 The radius of a semicircle is 2.7 m.Work out its area.

6 The diameter of a semicircle is 8.2 cm.Work out its area.

7 The diagram shows a running track.The ends are semicircles of diameter57.3 m and the straights are 110 m long.

Work out the area enclosed by the track.Give your answer correct to the nearest m2.

8 The diagram shows a circle of diameter6 cm inside a square of side 10 cm.

a Work out the area of the square.

b Work out the area of the circle.

c By subtraction, work out the area of the shaded part of the diagram.

9 The diagram shows a circle of radius 7 cminside a circle of radius 9 cm.

Work out the area of the shaded part ofthe diagram, correct to the nearest cm2.

10 The diagram shows an 8 cm by 6 cm rectangle inside a circle of diameter 10 cm.

Work out the area of the shaded part ofthe diagram.

22.3 Circumferences and areas in terms of �Answers to questions involving the circumference or area of a circle are sometimes givenin terms of �, which is exact, not as a number, which is approximate.

The diameter of a circle is 8 cm. Find the circumference of the circle.Give your answer as a multiple of �.

Solution 6� � 8

Circumference � 8� cm

Example 6

409

22.3 Circumferences and areas in terms of � CHAPTER 22

8.2 cm

57.3 m

110 m

6 cm

10 cm

10 cm

9 cm

7 cm

8 cm

6 cm10 cm

Multiply the diameter by �.

Write the 8 before the �. The units are cm.

2.7 m

410

CHAPTER 22 Circumference and area of a circle

The radius of a circle is 3 m.Find the area of the circle.Give your answer as a multiple of �.

Solution 7� � 32 � � � 9

Area � 9� m2

The diameter of a semicircle is 12 cm.Find the perimeter of the semicircle.Give your answer in terms of �.

Solution 8The perimeter is the sum of the arc length and the diameter.

�� �

2

12� � �

12

2

�� � 6�

Perimeter � 6� � 12 cm

If the circumference of a circle is given as a multiple of �, its diameter can be found.

The circumference of a circle is 30� m. Find its radius.

Solution 9

d � �30

�

�� � 30

�3

2

0� � 15

Radius � 15 m

Exercise 22C

In Questions 1–4, give the answers as multiples of �.

1 Find the circumference of a circle with a diameter of 7 m.

2 Find the area of a circle with a radius of 5 cm.

3 Find the circumference of a circle with a radius of 8 cm.

Example 9

Example 8

Example 7

Square the radius and then multiply by �.

Write the 9 before the �. The units are m2.

The arc length is half the circumference of a circle with adiameter of 12 cm.

To find the perimeter, add the diameter and the arclength. The units are cm.

To find the diameter, divide the circumference by �.

To find the radius, divide the diameter by 2.The units are m.

12 cm

411

Chapter summary CHAPTER 22

4 Find the area of a circle with a diameter of 20 m.

5 The diameter of a semicircle is 18 cm. Find the perimeter of the semicircle.Give your answer in terms of �.

6 The radius of a semicircle is 7 cm. Find the perimeter of the semicircle.Give your answer in terms of �.

7 The radius of a semicircle is 10 cm. Find its area. Give your answer as a multiple of �.

8 The circumference of a circle is 16� cm. Find its diameter.

9 The circumference of a circle is 60� m. Find its radius.

10 The circumference of a circle is 14� cm. Find its area. Give your answer as a multipleof �.

Chapter summary

Chapter 22 review questionsIf your calculator does not have a � button, take the value of � to be 3.142. Giveanswers correct to 1 decimal place, unless the question states otherwise.

1 Work out the circumference of a circle with a diameter of 27 cm.

2 Work out the circumference of a circle with a radius of 8.7 m.

3 Work out the diameter of a circle with a circumference of 24.7 cm.

You should now know:

how to find the circumference of a circle using

circumference of a circle � � � diameter

the formulae C � �d and C � 2�r

how to find the diameter (or radius) of a circle if its circumference is known using

diameter ��circum

�

ference� or using the formula d � �

C

��

how to find the area of a circle using

area of a circle � � � radius � radius

the formula A � �r2

how to solve problems involving the circumference and area of a circle, includingcompound shapes and shaded areas

how to express answers to questions involving the circumference or area of a circle interms of �.

�

�

�

�

�

�

4 Work out the radius of a circle with a circumference of 53.2 cm.

5 Work out the area of a circle with a radius of 7.9 m, correct to the nearest m2.

6 Work out the area of a circle with a diameter of 3.2 cm.

7 Stonehenge is surrounded by a circular ditch with a diameter of 104 m.Work out the total distance round the ditch.Give your answer correct to the nearest metre.

8 A discus thrower’s circle has a diameter of 2.5 m.Work out the area of the circle.

9 Work out the perimeter of a semicircle with a diameter of 11 cm.

10 Work out the area of a semicircle with a radius of 7.4 m.

11 Find the circumference of a circle with a diameter of 9 cm. Give your answer as amultiple of �.

12 Find the area of a circle with a radius of 4 cm. Give your answer as a multiple of �.

13 A table has a top in the shape of a circle with a radius of 45 centimetres.

a Calculate the area of the circular table top.Give your answer correct to the nearest cm2.

The base of the table is also in the shape of a circle.The circumference of this circle is 110 centimetres.

b Calculate the diameter of the base of the table.

(1384 Nov 1996)

14 The diagram shows a shape, made from a semicircle and a rectangle.The diameter of the semicircle is 12 cm.The length of the rectangle is 14 cm.

Calculate the perimeter of the shape.Give your answer correct to 1 decimal place.

(1385 June 2002)

412

CHAPTER 22 Circumference and area of a circle

12 cm

14 cm Diagram NOTaccurately drawn

15 A mat is made in the shape of a rectangle with a semicircle added atone end.The width of the mat is 1.52 metres.The length of the mat is 1.86 metres.

Calculate the area of the mat.Give your answer in square metres,correct to 2 decimal places.

(1385 Nov 1999)

16 The diagram shows a shape.AB is an arc of a circle, centre O.Angle AOB � 90°.OA � OB � 6 cm.

Calculate

a the area

b the perimeter of the shape.

17 The diagram shows a circle of diameter 70 cm inside a square of side 70 cm.Work out the area of the shaded part of the diagram.Give your answer correct to the nearest cm2.

(1384 Nov 1997)

18 The diagram shows a right-angled triangle ABC and a circle.A, B and C are points on the circumference of the circle.AC is a diameter of the circle.The radius of the circle is 10 cm.AB � 16 cm and BC � 12 cm.

Work out the area of the shaded part of the circle.Give your answer correct to the nearest cm2.

(1385 June 1999)

413

Chapter 22 review questions CHAPTER 22

1.52 m

1.86 m

Diagram NOTaccurately drawn

6 cm

6 cm

A

O B

Diagram NOTaccurately drawn

70 cm

70 cm

10 cm 10 cm

12 cm16 cm

A

B

C

Diagram NOTaccurately drawn

19

a Work out the area of Shape A.

b i Work out the perimeter of the semicircle, Shape B.ii Work out the area of the semicircle, Shape B.

(1385 June 1998)

20 The diagram shows a rectangle drawn inside a circle.The centre of the circle is at O.The rectangle is 15 cm long and 9 cm wide.

Calculate the circumference of the circle.

(1385 Nov 2001)

21 The diameter of a circle is 12 centimetres.

a Work out the circumference of the circle.Give your answer in centimetres correct to one decimal place.

The length of each diagonal of a square is 20 cm.

b Work out the area of the square.

(1387 Nov 2004)

414

CHAPTER 22 Circumference and area of a circle

4 cm

4 cm

1 cm

2 cm

2 cm1 cm

1 cm

3 cm

Shape A

Diagrams NOTaccurately drawn

Shape B

10 cm

15 cm

9 cmO

Diagram NOTaccurately drawn

12 cm

Diagram NOTaccurately drawn

Diagram NOTaccurately drawn