Circular Measure

Radian MeasureLength of ArcArea of Sector

Area of Segment

Radian Measure

rr

1 radian

Radian Measure

360° in a full rotation

Know how to convert from degrees to radians and vice versa

2π = 360° π = 180°

Examples Find the radian measure

equivalent of 210°. Find the degree measure equivalent

of radians.3π 4

°4

31804

3π180° = π

π180

°

210π 180

°

7π 6

r

Length of Arc

l

θ

θ must be in radians!

Fraction of circle

Length of arc

2

Circumference = 2πr 22

s r

s r

r

Area of Sector

2Area2

r

Fraction of circle

Area of sector

2

Area of circle = π r 2

212

r

θ

θ must be in radians

r

θ

θ must be in radians

21Area of segment sin2

r

θ

Examples

s = 2·5 8

s = rθ l

2·58 cm

A circle has radius length 8 cm. An angle of 2.5 radians is subtended by an arc. Find the length of the arc.

(i) Find the length of the minor arc pq.

(ii) Find the area of the minor sector opq.

p

qo

10 cm

0·8 rad

p

qo12 cm

56

s = rθ = 10(0·8)

212

A r 21 (10) (0·8)2

s = rθ

212

A r 21 5(12)2 6

5(12)6

Q1. Q2.

Q3. The bend on a running track is a semi-circle of radius

A runner, on the track, runs a distance of 20 metres on the bend. The angles through which the runner has run is A.

Find the measure of A in radians.

20 mA

100 π

metres.

20 = θ100 π

π 100 θ = 20

s = rθ

2·5 9

Q4. A bicycle chain passes around two circular cogged wheels. Their radii are 9 cm and 2·5 cm. If the larger wheel turns through 100 radians, through how many radians will the smaller one turn?

100 radians s = rθs = 9 100 = 900 cm

900 = 2·5θ

θ = 900 2·5

The diagram shows a sector circumscribed by a circle

k k

60º

r

r

30ºk2

r

k2 3

2

32

cos 30º

kr

31

3kr

(i) Find the radius of the circle in terms of k.

k2

rcos 30º

The diagram shows a sector circumscribed by a circle

k kr

r

(ii) Show that the circle encloses an area which is double that of the sector.

Area of circle π r2

3kr

π

2

3k

2

3k

Area of sector 212

r 212 3

k

2

6k

Twice area of sector