Question 6
(USS) A convex planar polygon $\mathcal M$ with perimeter $l$ and area $S$ is given. Let $M(R)$ be the set of all points in space that lie a distance at most $R$ from a point of $\mathcal M$. Show that the volume $V(R)$ of this set equals
$$V(R)=\frac43\pi R^3+\frac\pi2 lR^2+2SR.$$
$$V(R)=\frac43\pi R^3+\frac\pi2 lR^2+2SR.$$