Question

A circle has a chord that goes from `pi/3` to `pi/8` radians on the circle. If the area of the circle is `48 pi`, what is the length of the chord?

Answer 1

The chord, `c ~~ 4.45`

Explanation:

Given the Area, A, of the circle is `48pi`, allows us to solve for the radius, using the equation:

`A = pir^2`

`48pi = pir^2`

`r^2 = 48`

Two radii and the chord form a triangle with the angle, `theta = pi/8 - pi/3`, between the two radii. This allows us to use the Law of Cosines to find the length of the chord, c:

`c^2 = r^2 + r^2 - 2(r)(r)cos(theta)`

`c^2 = r^2 + r^2 - 2(r^2)cos(theta)`

`c^2 = 2r^2(1 - cos(theta))`

`c^2 = 96(1 - cos(pi/8 - pi/3))`

`c = sqrt(96(1 - cos(pi/8 - pi/3)))`

`c ~~ 4.45`