Euler Polyhedron Formula

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


 * $$V - E + F = 2$$.

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.