# Euler Polyhedron Formula

## Contents

## Theorem

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

- $V - E + F = 2$

## Proof

Let $G$ be a planar graph with $V$ vertices and $E$ edges.

Let $V = 1$.

The number of faces $F$ is then given by:

- $F = E + 1$

Hence:

- $V - E + F = 1 - E + \left({E + 1}\right) = 2$

and the formula holds.

Otherwise, let $G$ be any (planar) 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 $G$ was the same as the one considered in the previous case.

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

From Polyhedra and Plane Graphs, any polyhedron's vertices, edges and faces may be represented by a (planar) graph.

Hence the formula applies to polyhedra.

$\blacksquare$

## Source of Name

This entry was named for Leonhard Paul Euler.

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 3$: Appendix $\text A$: Euler - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $2$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.21$: Euler ($1707$ – $1783$) - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2$