Circular MeasureArc Length and Area of a Sector

Review: Radian MeasureIn general, if the length of arc, s units

and the radius is r units, then

For example:

If s = 3 cm and r = 2 cm, then

That is the size of the angle (θ) is given by the ratio of the arc length to the length of the radius.

Arc LengthFrom our definition of the radian, we have:

where θ is in radiansFor example:

If θ = 2.1 radians and r = 3 cmLength of arc AB, s = rθ= 3 × 2.1 cm= 6.3 cm

s = rθ

Example 1The diagram shows part of a circle, centre O, radius r cm. Calculate:

(a)The value of r,( )b BOC in radians.

P

C

A

B

O

1.2 rad

(a)In the sector AOB, s = 2.4 cm and θ = 1.2 radians (b)In the sector BOC, s = 1.4

cm and r = 2 cm

Area of a Sector

r

r

A s

In the diagram, the angle of the sector AOB is θ radians.By proportion:Let the area of sector AOB be A.Thus,

Now, as s = rθ, we have:

Example 2In the diagram, arc Ab and CD are arc of concentric circles, centre ). If OA = 6 cm, AC = 3 cm and the area of sector AOB is 12 cm2, calculate

( )a AOB in radians,(b)The area and perimeter of the shaded

region.

6 cm 3 cm

D

CrO A

B

(a)Let be AOB = θ radiansArea of the sector AOB = 12 cm2(b) Now OC = OA + AC = 9 cm

Area of the shaded region = area of sector COD – area of sector AOB

= 27 – 12 = 15 cm2

Arc Length and Area of a Sector

where θ is in radians

Connection: Area of Trianglewhere C is an acute angle

Area of Trianglewhere C is an obtuse angle

Example 3

Solution:

Problem

Solutions