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

Carter and Wagon's proof without words:

enter image description here

(Image author: Christian Lawson-Perfect, source)


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: enter image description here

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

Tags:

Geometry

Area

Algebra Precalculus

Circles

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