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