The Perimeter of a Polyomino and the Surface Area of a Polycube

A polyomino is the union of a collection of unit tiles that meet edge-t0-edge and are connected. A polyomino has no holes, so it’s boundary is a single polygon. A natural question is whether the perimeter of the polyomino can be determined from the number of tiles and some other elementary parameter.

I will be proving the following Theorem: The perimeter of a polyomino with n tiles and r interior vertices is 2n – 2r +2, as well as Pick’s Theorem.