Post on 24-Dec-2015
Area of a Circular Segment
Objectives:
•Review Area of Circles & Sectors
•Find the Area of a Circular Segment
Anthony E. Davis
Summer 2003
Area of a Circle
Area r 2
radius
Area of a Sector
A sector is a pie-like shape of the circle, where theta is the measure of the central angle.
(a percentage of the circle)
Area
360
r
2
radius
radius
Area of a Segment
A segment is the portion of the circle with the radii connected at
their endpoints and the central arc.
(The crust of the pie slice.)
radius
radius
The red piece is the segment.
Area of a Segment
The segment is the sector minus the triangle.
Area of a sector is .
The area of the triangle is
(since we know two sides and the included angle).
Hence the area of a segment is :
Area
360
r
2
Area 1
2r 2 sin
Area
360
r
2 1
2r 2 sin
Example
Find the area of a segment of a circle whose diameter is 20 cm and central angle is 29 degrees.
Area
360
r
2 1
2r 2 sin
Area 29360
(10)2
1
2(10)2 sin 29
Area 1.07cm2
Practice #1Find the area of a circular segment to the nearest
tenth, if the central angle is 85 degrees with a radius of 2.1 ft.
2 2
2 2
2
1sin
360 2
85 1(2.1) (2.1) sin85
360 2
1.07
Area r r
Area
Area ft
Practice #2Find the area of a circular segment to the nearest tenth,
if its central angle is 5π/8 with a radius of 6 cm.
58
360
2112.5
2 2
2 2
2
1sin
360 2
112.5 1(6) (6) sin112.5
360 2
18.71
Area r r
Area
Area cm
Practice #3
If the area of a circular segment is 54 ft. squared
and contains a central angle of 30 degrees, what is the value of the radius
to the nearest tenth?
2 2
2 2
2 2
2 2
2
2
1sin
360 2
30 154 ( ) ( ) sin 30
360 2
1 154 sin 30
12 2
648 6 sin 30
648 6sin 30
648
6sin 30
648
6sin 30
67.65
Area r r
r r
r r
r r
r
r
r
ft r