The Lune of Hippocrates
In the diagram below, which has the larger area, the triangle or the crescent?
Examine the diagram carefully, and you'll see that it is composed of a circular sector and a semicircle.
Let the radius of the sector be 1. Then the diameter of the semicircle is Root(2), so the area of both must be PI/4.
Excluding the area shared by both, therefore, the area of the triangle and the crescent shape is the same.
