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This presentation shows how to find the arc length and area of sector of a circle
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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