AREA AND CIRCUMFERENCE OF A CIRCLE

diam

eter

radius

circumference

The perimeter of a circle is called the circumference (C).

The diameter (d) of a circle is twice the radius (r).

C 2r or C d

3.142

rearranging the sectors of the circle gives

Consider dividing a circle into 8 equal sectors:

Now consider dividing the circle into 12 equal sectors:

rearranging the sectors of the circle gives

The more sectors there are, the closer the area gets to being a rectangle.

r

rhalf circumference

Area r 2

Examples1 Calculate a the circumference b the area of the circle.

8 cm

a Circumference d

8

25.1 cm (to 3 s.f.)

b Area r 2

42

50.3 cm2 (to 3 s.f.)

16

Examples2 Calculate a the circumference b the area of the circle.

3.7 cm

a Circumference d

7.4

23.2 cm (to 3 s.f.)

b Area r 2

3.72

43.0 cm2 (to 3 s.f.)

13.69

Examples3 The radius of the circle is 5 cm. The circle touches all four sides of the square. Calculate the shaded area.

Shaded area area of square area of circle

10 10 52

21.5 cm2 (to 3 s.f.)

100 25

10 cm

Examples4 The radius of the large circle is 8 cm. The radius of the small circle is 5 cm. Calculate the shaded area.

Shaded area area of large circle area of small circle

82 52

126 cm2 (to 3 s.f.)

64 25

39

Examples5 The radius of the circle is 3 cm. The vertices of the square lie on the circumference of the circle. Calculate the shaded area.

Let length of sides of square x

x2 x2 62

x 4.243... cm

2x2 366 cm

Shaded area area of circle area of square

32 4.243 4.243

10.3 cm2 (to 3 s.f.)

9 18

x2 18

x

x

Using Pythagoras

Examples6 The area of the circle is 200cm2. Find the value of r.

r cm Area r 2

200 r 2

r 63.66

r 2

200

r2 63.66

r 7.98 cm (to 3 s.f.)

Semicircles and quadrants

A semicircle is half a circle. A quadrant is quarter of a circle.

Examples1 Calculate a the perimeter b the area of the semicircle.

10 cm

a Perimeter

d

210

10

210

25.7 cm (to 3 s.f.)

b Area

r 2

2

52

2

39.3 cm2 (to 3 s.f.)

Examples2 Calculate a the perimeter b the area of the quadrant.

9 cm

a Perimeter

d

4 9 9

18

218

46.3 cm (to 3 s.f.)

b Area

r 2

4

92

4

63.6 cm2 (to 3 s.f.)

Examples3 The perimeter of the quadrant is 50 cm. Calculate the value of r.

r cm

Perimeter

2r

4 r r

50

r

2 2r

50 1.571r 2r

50 3.571r

r 14.0 cm (to 3 s.f.)

r

50

3.571

Examples4 The diagram shows four identical circles of radius 4 cm. Calculate the shaded area.

Shaded area area of square area of 4 quadrants

8 8 42

13.7 cm2 (to 3 s.f.)

64 16

area of square area of circle

Arcs and sectors

An arc is part of the circumference of a circle.

A sector of a circle is a region bounded by an arc and two radii.

r

r

sector

arc

Arc length

360

2r

Area of sector

360

r 2

95o 5 cm

5 cm

O

A

B

Examples1 Find a the area of sector OAB

b the length of arc ABc the perimeter of sector OAB.

a Area of sector

95

360 52

20.7 cm2 (to 3 s.f.)

b Arc length

95

3602 5

8.2903...

8.29 cm (to 3 s.f.)

c Perimeter OA OB arc AB

5 5 8.2903...

18.3 cm (to 3 s.f.)

Examples2 Find a the area

b the perimeter of the shape.

7 m 3 m

a Area

126

360 102

126

360 72

56.1 m2 (to 3 s.f.)

b Perimeter large arc small arc 3 3

21.99 15.34 3 3

43.4 cm (to 3 s.f.)

126° 110.0 53.88

126

3602 10

126

3602 7

3 3