Euler's polyhedron formula

Statement

Let $P$ be a convex polyhedron in $\mathbb{R}^3$ with $V$ vertices, $E$ edges and $F$ faces. Then $$V - E + F = 2.$$

Remarks

The number $V - E + F$ is the Euler characteristic of the surface of $P$. The value $2$ is the Euler characteristic of the sphere. The formula also holds for each polyhedron whose surface is topologically a sphere and whose faces are topologically discs. It can fail for other polyhedra. The surface of a polyhedron with the shape of a torus has $V - E + F = 0$. For the cube, $8 - 12 + 6 = 2$. For the regular tetrahedron, $4 - 6 + 4 = 2$.

History

Euler stated the formula in 1758. Cauchy gave a proof in 1811.

Explanations

L1 human

Count the corners, the edges and the faces of any convex solid with flat faces. Corners minus edges plus faces always gives $2$. Cut off one corner of a cube: you get $2$ more corners, $3$ more edges and $1$ more face, and the count stays $2$.