A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown. What is the area of the shaded region? $\textbf{(A)}\ 27\sqrt{3}-9\pi\qquad$ $\textbf{(B)}\ 27\sqrt{3}-6\pi\qquad$ $\textbf{(C)}\ 54\sqrt{3}-18\pi\qquad$ $\textbf{(D)}\ 54\sqrt{3}-12\pi\qquad$ $\textbf{(E)}\ 108\sqrt{3}-9\pi$
Solution
The hexagon has area $6\cdot \frac{\sqrt3}4\cdot 6^2=54\sqrt3$.
Each interior angle of the hexagon is $120^\circ$, so the six sectors have total angle $6\cdot120^\circ=720^\circ$, i.e. they form $2$ full circles with area $2\cdot \pi \cdot 3^2=18\pi$.
A regular octagon has side length $4$. At each vertex, a circular sector of radius $2$ is drawn inside the octagon, with its radii along the adjacent sides. What is the area inside the octagon but outside the eight sectors?
$\textbf{(A)}\ 27\sqrt{3}-9\pi\qquad$ $\textbf{(B)}\ 27\sqrt{3}-6\pi\qquad$ $\textbf{(C)}\ 54\sqrt{3}-18\pi\qquad$ $\textbf{(D)}\ 54\sqrt{3}-12\pi\qquad$ $\textbf{(E)}\ 108\sqrt{3}-9\pi$ 
