2008 AMC 12B Problem 15

On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let R be the region formed by the union of the square and all the triangles, and S be the smallest convex polygon that contains R. What is the area of the region that is inside S but outside R?

\textbf{(A)} \; \frac {1}{4} \qquad \textbf{(B)} \; \frac {\sqrt {2}}{4} \qquad \textbf{(C)} \; 1 \qquad \textbf{(D)} \; \sqrt {3} \qquad \textbf{(E)} \; 2 \sqrt {3}

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