An equilateral triangle has side length 6. What is the area of the region containing all points that are outside the triangle but not more than 3 units from a point on the triangle? $\mathrm{(A)}\ 36+24\sqrt{3}\qquad$ $\mathrm{(B)}\ 54+9\pi\qquad$ $\mathrm{(C)}\ 54+18\sqrt{3}+6\pi\qquad$ $\mathrm{(D)}\ \left(2\sqrt{3}+3\right)^2\pi\qquad$ $\mathrm{(E)}\ 9\left(\sqrt{3}+1\right)^2\pi$
Solution
The region consists of $3$ rectangles, each $3$ by $6$, one on each side, together with $3$ circular sectors of radius $3$ at the vertices. Each sector has angle $360^\circ-90^\circ-90^\circ-60^\circ=120^\circ$, so together they form a whole circle of radius $3$.
An equilateral triangle has side length $8$. What is the area of the region containing all points that are outside the triangle but not more than $2$ units from a point on the triangle?
