2001 Cayley Problem 20

Square ABCD with side length 2 is inscribed in a circle, as shown. Using each side of the square as a diameter, semi-circular arcs are drawn. The area of the shaded region outside the circle and inside the semi-circles is

\textbf{(A)}\ \pi\quad \textbf{(B)}\ 4\quad \textbf{(C)}\ 2\pi-2\quad \textbf{(D)}\ \pi+1\quad \textbf{(E)}\ 2\pi-4

If there are no answer choices shown, enter a numerical answer.

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