1. Radian Measure Length of Arc Area of Sector
Circular Measure
Radian Measure Length of Arc Area of Sector
Area of Segment

2. Radian Measure r
1 radian

3. Radian Measure  360° in a full rotation
 Know how to convert from degrees to radians and vice versa
2π = 360°
π = 180°

4. Examples 180° = π = 4 3π 4 3180 π 180 1° = = 135° 210π 210° = 7π 6 =
 Find the degree measure equivalent
of radians. 3π 4
 Find the radian measure equivalent of 210°.
180° = π = 4 3π 4
3180 π
180
1° =
= 135°
210π
180
210° = 7π 6 =

5. Length of Arc l r θ θ must be in radians! Fraction of circle
Circumference = 2πr

6. Area of Sector r θ θ must be in radians Fraction of circle
Area of circle = π r 2

7. θ must be in radians r θ θ

8. Examples
 A circle has radius length 8 cm. An angle of 2.5 radians is subtended by an arc.
Find the length of the arc.
s = rθ l
2·5
s = 2·5  8
8 cm

9. (i) Find the length of the minor arc pq.
(ii) Find the area of the minor sector opq.
Q1.
Q2. p p
10 cm
12 cm o q o q
0·8 rad
s = rθ
= 10(0·8)
s = rθ

10. s = rθ 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.
100 π
metres.
20 m A
s = rθ
20 = θ
100 π
π 100
θ = 20 

11. s = rθ 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?
2·5 9
s = rθ
100 radians
s = 9  100 = 900 cm
900 = 2·5θ
θ =
900
2·5

12. The diagram shows a sector circumscribed by a circle
(i) Find the radius of the circle in terms of k. k 2
60º k 2 r
cos 30º = 3 2
cos 30º =
30º r k k r k 2 3 = r k r 3 1 = 3 k
r =

13. The diagram shows a sector circumscribed by a circle
(ii) Show that the circle encloses an area which
is double that of the sector. p 3 3 k
r = r k k
Area of circle = π r2
æ ö
ç ÷
è ø
= π 2 3 k r
Twice area of sector
Area of sector