IMO 1966 LL USS6

Origin: USS

Problem

A convex planar polygon 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 M. Show that the volume V (R) of this set equals V (R) = 4 3\piR3 + \pi 2 lR2 + 2SR.