2007 Cayley Problem 23

In the diagram, ABCD is rectangle with AB = 12 and BC = 18. Rectangle AEFG is formed by rotating ABCD about A through an angle of 30^\circ. The total area of the shaded regions is closest to

\textbf{(A)}\ 202.8\quad \textbf{(B)}\ 203.1\quad \textbf{(C)}\ 203.4\quad \textbf{(D)}\ 203.7\quad \textbf{(E)}\ 204.0

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Cayley contests here.