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? $\textbf{(A)}\ 1 - \frac{\sqrt2}{2} + \frac{\pi}{4}\qquad\textbf{(B)}\ \frac{1}{2} + \frac{\pi}{4} \qquad\textbf{(C)}\ 2 - \sqrt2 + \frac{\pi}{4}$ $\textbf{(D)}\ \frac{\sqrt2}{2} + \frac{\pi}{4} \qquad\textbf{(E)}\ 1 + \frac{\sqrt2}{4} + \frac{\pi}{8}$
Solution
The four $45^\circ$ sectors have total area $\frac{\pi}4$.
The four right triangles with legs $\frac12$ have total area $\frac12$.
The four small isosceles right triangles with leg $\frac{\sqrt2}2-\frac12$ have total area $\frac32-\sqrt2$.
