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

Radian Measure r 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 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 =

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

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

θ must be in radians r θ θ

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

(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

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 

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

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 =

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