Euler's Formula for Polyhedra

What just happened?

You changed the solid — from the simple tetrahedron to the prism. Each time the numbers of vertices $V$, edges $E$ and faces $F$ were different, yet the combination $V - E + F$ always gave exactly 2. This is no accident — it is one of the most beautiful results in mathematics.

Euler's formula for polyhedra:

For every convex polyhedron, the number of vertices ($V$) minus the number of edges ($E$) plus the number of faces ($F$) equals 2 — no matter how complicated the solid is.

Check it yourself:

Cube: $8 - 12 + 6 = 2$. Tetrahedron: $4 - 6 + 4 = 2$. Octahedron: $6 - 12 + 8 = 2$. It always comes out 2.

Where did it come from?

The relation was discovered by Leonhard Euler — he announced it in a letter to Goldbach in 1750 and published it in 1752. It links three seemingly independent features of a solid and holds from the simplest tetrahedron to solids with hundreds of faces.

The invariant — the Euler characteristic:

The number $V - E + F$ is the Euler characteristic of a surface. For all solids equivalent to a sphere it equals 2, but for a solid with a "hole" (like a torus, i.e. a bagel) it equals 0. This number is invariant under continuous deformations of shape — which is why the formula gave rise to a new branch of mathematics: topology (the study of properties of shapes invariant under stretching and bending without tearing).

Why you need this for the exam:

The formula $V - E + F = 2$ lets you quickly find the missing value when you know the other two (e.g. the number of faces, given the vertices and edges). It is also a classic example of an invariant and an introduction to planar graphs. Applications reach into chemistry (fullerenes) and computational geometry.

Remember: for every convex polyhedron $V - E + F = 2$ — vertices minus edges plus faces always give 2.