We draw semicircles outward on the sides of a square as diameters and draw the circle circumscribed around the square. The area of the crescent moons left out together gives the area of the square.
The statement is true for any rectangle. Let $a$ and $b$ be the adjacent sides. The area of the two opposite semicircles is $(a / 2)^{2} \pi$ and $(b / 2)^{2} \pi$. The area of the rectangle is $a b$. The diameter of the circumscribed circle is $\sqrt{a^{2}+b^{2}}$, so its area is $\left(a^{2}+b^{2}\right) / 4 \pi$. Thus, the area of the crescents is $(a / 2)^{2} \pi+(b / 2)^{2} \pi+a b-\left[\left(a^{2}+b^{2}\right) / 4\right] \pi=a b$.