AMC 12 2008b 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 $R$?

(Solution)

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