Euler Polyhedron Formula

Theorem

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

$V - E + F = 2$

Examples

Tetrahedron

The tetrahedron has:

$4$ vertices
$6$ edges
$4$ faces.

We see that:

 $\displaystyle V - E + F$ $=$ $\displaystyle 4 - 6 + 4$ $\displaystyle$ $=$ $\displaystyle 2$

and so the Euler Polyhedron Formula is seen to hold.

Cube

The cube has:

$8$ vertices
$12$ edges
$6$ faces.

We see that:

 $\displaystyle V - E + F$ $=$ $\displaystyle 8 - 12 + 6$ $\displaystyle$ $=$ $\displaystyle 2$

and so the Euler Polyhedron Formula is seen to hold.

Source of Name

This entry was named for Leonhard Paul Euler.