How can i Prove that the gray area is the same as white area?

Question

A circle is cut into 8 parts, each part has the angle 45 degrees from an arbitrary point. how to prove that the white area is the same as the Gray area?

I just want any hint for solving this question. how can I prove this?

Answer 1

Carter and Wagon's proof without words:

(Image author: Christian Lawson-Perfect, source)

Answer 2

This theorem is known as the Pizza theorem (see https://en.wikipedia.org/wiki/Pizza_theorem for further reading). There exist algebraic proofs, but the most elegant proof for the 8 segment case is below:

In general, alternating areas are equal iff the number of segments is a multiple of 4 at least 8.