Question

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

Answer 1

\[\huge{V(R)=\frac{ 4 }{ 3 }\pi R^3+\frac{\pi l}{2} R^3+2SR}\]