Euler Polyhedron Formula

Theorem

For any convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces:

$V - E + F = 2$.

Proofs

Induction on Vertices

Let $G$ be a graph with one vertex and $E$ edges.

The faces then number $F = E + 1$, hence $V - E + F = 1 - E + (E+1) = 2$ and the formula obtains.

Otherwise, let $G$ be any graph.

Since contracting any edge decreases the number of vertices and edges each by one, the value of $V - E + F$ remains unchanged.

Hence by induction through contracting edges indefinitely, the value remains the same as if the graph was the same as the one considered in the previous case.

Hence $V - E + F = 2$ for any graph.

From Polyhedra and Plane Graphs, any polyhedron's vertices, edges, and faces may be represented by a graph, so the formula applies to polyhedra.

$\blacksquare$