Question

A chord with a length of `24` runs from `pi/3` to `pi/2` radians on a circle. What is the area of the circle?

Answer 1

The area of the circle is `6753.36`

Explanation:

The chord forms the base of of an isosceles triangle where the legs are the radii

The internal angles of a triangle sum to `Pi`
The base angles of an isosceles triangle, where the apex is `a` radians can be calculated as
`(Pi - a)/2`

The angle at the apex of the triangle is `Pi/6`
Therefore, the equal angles are `(Pi - Pi/6)/2` or `5Pi/12`radians

The leg of an isosceles triangle can be calculated as half the base divided by the cosine of the base angle

Therefore, the radius (`r`) can now be calculated as
`r = 24/2 * 1/cos(5Pi/12) = 46.36444`

The area of a circle is given by `Pir^2`

The area of the circle is `Pi*46.36444*46.36444 = 6753.36`