AMC 8 2011 Problem 25

A circle with radius $1$ is inscribed in a square and circumscribed about another square as shown. Which fraction is closest to the ratio of the circle's shaded area to the area between the two squares?

$\textbf{(A)}\ \frac{1}2 \textbf{(B)}\ 1 \textbf{(C)}\ \frac{3}2 \textbf{(D)}\ 2 \textbf{(E)}\ \frac{5}2$