Question

Express answer in exact form.

An equilateral triangle is inscribed in a circle with a radius of 6". Find the area of the segment cut off by one side of the triangle.

help me please..

Answer 1

The correct way to do this is:

The inscribed triangle can be cut into 6 equal triangles. One side each of these triangles (the hypotenuse) is equal to the length of the radius. Each triangle is a rt. triangle. Each triangle is a 30 - 60 - 90 triangle.

r = 6
So, the sides of the triangle are:
Remember for a 30-60-90 triangle: 1 - 2 (hypotenuse) - √3
3 - 6 - 3√3

Area of one triangle:
A = (1/2)bh
b = 3
h = 3√3

A = (1/2)(3)(3√3) = 9√3/2

6A = 6(9√3/2) = 27√3

Area of circle:
A = pi 6^2

A = 36pi

Area of segment (there are 3):
A = (area of circle - area of triangle)/3

A = (36pi - 27√3)/3 = 12pi - 9√3

12pi - 9√3 is the exact answer.

BTW, I got this off of yahoo, so the creds don't go to me.