God! This is going to be awfully hard to explain in a rigorous manner, but I'll take a whack at it. First of all, let's notice what $M(R)$ "looks like". If we place spheres of radius $R$ centered at each of the vertices of the polygon, then $M(R)$ is precisely the convex hull of the spheres. Alternatively, you may think that we place a sheet of rubber around the whole thing (polygon and spheres), and the set is what we obtain after the sheet has shrinked as much as possible. a) In order to compute the volume, we need to compute the volume of three parts: the prism having the polygon as the median plane and having a height of $2R$ (this has volume $S\cdot R+S\cdot R=(2\cdot S)R$), the half-cylinders around the edges (this part has volume $\frac 12\sum a_i\cdot \pi R^2=(\frac \pi 2\cdot l)R^2$, where $a_i$ are the sides of the polygon), and whatever remains around the vertices. This last part is made up of sphere slices, which we can slide around on the sides of the polygon so that they form a sphere of radius $R$, so the volume is $(\frac 43\cdot \pi)R^3$. b) The area is, again the sum of the areas of the three regions we have defined above: the area of the bases of the prism ($2S$), the areas of the half-cylinders, which is $l\cdot \pi R$, and the area of a sphere of radius $R$, which is $4\pi R^2$. The total area is thus $4\pi R^2+(\pi\cdot l)R+2\cdot S$. c) Something similar to a). I'll be back later to edit this.