Find the area of a segment formed by a chord 8" long in a circle with radius of 8"

Question

This is a geometry problem. in the section on area of a circle.

 

Answer 1

draw 2 radii such that their endpoints on the circle are 8 inches apart

connect these points; this is the chord and it is 8 inches long

the 2 radii and the chord form an equilateral triangle

each side is 8 inches and each angle is 60 degrees

find the area of the sector formed by the 2 radii

find the area of the equilateral triangle

subtract the area of the triangle from the area of the sector and you have the area of the segment of the circle

area of the sector: A=(q/360)*∏r²

A=(60/360)*∏*8²

A=(1/6)(∏)(64)

A=(64/6)(∏)

A=(32/3)(3.14159)

A=33.5103 square inches

area of the triangle:

A=(√3/4)a² where a=length of a side

A=(1.73205/4)(8²)

A=(1.73205/4)(64)

A=(1.73205)(16)

A=27.7128 square inches

subtract

33.5103-27.7128=5.7975 square inches

the area of the segment is 5.7975 square inches