A cute geometry problem

I came across a cute geometry problem recently and I would like to pass it along.

Problem

Problem Statement

If the sides of the square are of unit length and all curves are circular arcs, what is the area of the highlighted region?

Although substantially easier with the use of calculus or trigonometry, this problem can be solved entirely with basic geometry (no weird laws you might have forgotten since high school are necessary).

I have derived a geometric solution, which follows, but I highly recommend trying to do it yourself first.

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