Euler Polyhedron Formula

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Theorem

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

$V - E + F = 2$


Proof


This theorem requires a proof.
In particular: There should be a proof that the net of the polyhedron is a planar graph. The result then follows from Euler's Theorem for Planar Graphs.
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Also known as

The Euler Polyhedron Formula is also known as Euler's theorem (for polyhedra).


Examples

Tetrahedron

The tetrahedron has:

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

We see that:

\(\ds V - E + F\) \(=\) \(\ds 4 - 6 + 4\)
\(\ds \) \(=\) \(\ds 2\)

and so the Euler Polyhedron Formula is seen to hold.


Cube

The cube has:

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

We see that:

\(\ds V - E + F\) \(=\) \(\ds 8 - 12 + 6\)
\(\ds \) \(=\) \(\ds 2\)

and so the Euler Polyhedron Formula is seen to hold.


Also see


Source of Name

This entry was named for Leonhard Paul Euler.


Sources

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