Euler's Formula

The sum of the number of faces (F) and vertices (V) of a polyhedron is two more than the number of its edges (E).

As an example, this can be verified for the following polyhedron.

The above polyhedron has 7 faces, 10 vertices, and 15 edges.

Euler's formula takes a slightly different form in two dimensions.

Here, F is the number of regions formed by V vertices, linked by E segments. This formula can be verified for the net of the previous polyhedron.

The net shown has 7 regions, 18 vertices, and 24 segments.