Question

A circle has a chord that goes from `( pi)/6` to `(5 pi) / 6` radians on the circle. If the area of the circle is `18 pi`, what is the length of the chord?

Answer 1

`c = sqrt(54)`

Explanation:

We can compute the radius from the area of the circle:

`pir^2=18pi`

`r = sqrt18`

To radii and the chord form a triangle. The angle between the two radii is:

`theta = (5pi)/6-pi/6= (2pi)/3`

If we use the angle and the length of the two radii, we can use the Law of Cosines:

`c^2=a^2+b^2-2(a)(b)cos(theta)`

where `a = b = r = sqrt18 and theta = (2pi)/3`

`c = sqrt((sqrt18)^2+(sqrt18)^2-2(sqrt18)(sqrt18)cos((2pi)/3)`

`c = sqrt(54)`