AMC 12 2012a Problem 14

The closed curve in the figure is made up of $9$ congruent circular arcs each of length $\frac{2\pi}{3}$, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side $2$. What is the area enclosed by the curve?

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label("$\circ$",(-0.865,0.5));
label("$\circ$",(0.865,-0.5));
label("$\circ$",(-0.865,-0.5));
label("$\circ$",(0,-1));
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draw(Arc((0.865,0.5),0.5,150,270));
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draw(Arc((-0.865,0.5),0.5,-90,30));

\textbf{(A)}\ 2\pi+6 \textbf{(B)}\ 2\pi+4\sqrt3 \textbf{(C)}\ 3\pi+4 \textbf{(D)}\ 2\pi+3\sqrt3+2 \textbf{(E)}\ \pi+6\sqrt3