Euler Polyhedron Formula

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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$6$ edges

$4$ faces.

We see that:

$\ds V - E + F$

$=$

$\ds 4 - 6 + 4$

$\ds $

$=$

$\ds 2$

and so the 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 is seen to hold.

Also see

Definition:Euler Characteristic of Surface

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 ($\text {1707}$ – $\text {1783}$)
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $2$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Euler's theorem: 1. (for polyhedra)
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Euler characteristic
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Euler's theorem: 1. (for polyhedra)
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Euler's Theorem
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): polyhedron (polyhedra)