2013 AMC 10A Problem 20

A unit square is rotated 45^\circ about its center. What is the area of the region swept out by the interior of the square?

\textbf{(A)}\ 1 - \frac{\sqrt2}{2} + \frac{\pi}{4}\qquad\textbf{(B)}\ \frac{1}{2} + \frac{\pi}{4} \qquad\textbf{(C)}\ 2 - \sqrt2 + \frac{\pi}{4}

\textbf{(D)}\ \frac{\sqrt2}{2} + \frac{\pi}{4} \qquad\textbf{(E)}\ 1 + \frac{\sqrt2}{4} + \frac{\pi}{8}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.