gwax's rumblings

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

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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