Euler formula

In a convex polyhedron, number of faces F, number of vertices S and number of edges A are related by :

  S+F=A+2  

For instance a cube has 6 faces, 8 vertices and 12 edges : 8+6=12+2

Proof

By induction. Let's consider an opened convex polyhedral area with a border being a (plane or not) polygonal line.
The relation to prove becomes : S+F=A+1.
This formula is true in the case of only one face (a polygon) where F=1 and S=A.
Adding a polygon with m sides and m vertices, touching the area border at p common edges, the result will have p+1 common vertices and F becomes F'=F+1, S becomes S'=S+m-(p+1) and A becomes A'=A+m-p,
Hence F'+S'-A'=F+S-A=1.
We can then end the proof by closing the polyhedron by a last face giving F'=F+1, A'=A and S'=S that is what we wanted.

Applications

Polyhedrons with holes

The relation is no more true for Polyhedrons with holes, or made of several connected parts.

 

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