An [equilateral triangle](https://artofproblemsolving.com/wiki/index.php/Equilateral_triangle) has side length $6$. What is the [area](https://artofproblemsolving.com/wiki/index.php/Area) of the region containing all points that are outside the triangle but not more than $3$ units from a point of the triangle?
$\mathrm{(A)}\ 36+24\sqrt{3}\qquad\mathrm{(B)}\ 54+9\pi\qquad\mathrm{(C)}\ 54+18\sqrt{3}+6\pi\qquad\mathrm{(D)}\ \left(2\sqrt{3}+3\right)^2\pi\\\mathrm{(E)}\ 9\left(\sqrt{3}+1\right)^2\pi$
The region described contains three rectangles of dimensions $3 \times 6$, and three $120^{\circ}$ degree arcs of circles of [radius](https://artofproblemsolving.com/wiki/index.php/Radius) $3$. Thus the answer is \[3(3 \times 6) + 3 \left( \frac{120^{\circ}}{360^{\circ}} \times 3^2 \pi\right) = 54 + 9\pi \Longrightarrow \mathrm{(B)}.\]