Euler's formula

In $1750$ the Swiss mathematician Leonhard Euler noticed something astonishing: for every polyhedron with no holes, the numbers of faces, edges, and vertices are connected by a single equation:

$F + V - E = 2.$

It works on a cube, a pyramid, a soccer-ball-shaped polyhedron , anything you can imagine, as long as it has flat faces and no doughnut-style holes.

Concept

The formula. For any convex polyhedron:

$F + V - E = 2.$

Some examples.

Why $2$? A proof is beyond class VIII, but the intuition is that you can flatten any polyhedron (project it onto a plane) and turn its faces, vertices, and edges into a planar graph. For any connected planar graph, $F + V - E = 2$ (here $F$ counts the regions of the plane including the outer "infinite" region).

Using it. If you know two of $F, V, E$, you can compute the third.

Example: a polyhedron has $20$ faces, $12$ vertices. How many edges?

• $E = F + V - 2 = 20 + 12 - 2 = 30$.

A constraint on Platonic solids. Suppose a regular polyhedron has $p$-sided faces, $q$ meeting at each vertex. Counting edges two ways:

• Each face has $p$ edges, each edge shared by $2$ faces: $E = \dfrac{pF}{2}$.
• Each vertex has $q$ edges, each edge has $2$ endpoints: $E = \dfrac{qV}{2}$.

Plugging into Euler's formula yields constraints on $p, q$ that allow only finite solutions , exactly the five Platonic solids. This is why there are only five.

What about doughnut shapes? A torus (a doughnut) is not a polyhedron, but if you triangulate its surface, you find $F + V - E = 0$. For shapes with $n$ holes, $F + V - E = 2 - 2n$. This generalisation is the start of topology.

Worked examples

Example 1. A polyhedron has $7$ faces and $15$ edges. Find $V$.

• $V = E + 2 - F = 15 + 2 - 7 = 10$.

Example 2. Can a polyhedron have $10$ faces, $20$ vertices and $20$ edges?

• Check: $10 + 20 - 20 = 10 \ne 2$. No, this is not a polyhedron.

Example 3. A prism with a $7$-sided base. Find $F, V, E$.

• $F = 7 + 2 = 9$ (sides + two bases).
• $V = 14$ (seven per base).
• $E = 21$ ($7 + 7 + 7$).
• Check: $9 + 14 - 21 = 2$. ✓

Example 4. A pyramid with $n$-sided base. Find $F, V, E$.

• $F = n + 1$, $V = n + 1$, $E = 2n$.
• $F + V - E = (n+1) + (n+1) - 2n = 2$. ✓ (Always holds.)

Try it yourself

1. A polyhedron has $V = 10, E = 15$. Find $F$.
2. A pentagonal prism: count $F, V, E$ and verify the formula.
3. A hexagonal pyramid: count $F, V, E$ and verify.
4. Can a polyhedron have $F=5, V=6, E=9$?
5. Octahedron has $8$ triangular faces. Verify it has $12$ edges using $E = pF/2$.
6. An icosahedron has $20$ triangular faces. How many vertices and edges?
7. Why are there only $5$ Platonic solids? Explain in one sentence.
8. A polyhedron has all square faces with $4$ meeting at each vertex. Identify it.

Activity

Build and verify. Make small polyhedra using straws and pipe cleaners (or sticks and modelling clay). Build a cube, a tetrahedron, and a square pyramid. Count $F, V, E$ for each by touching them. Note that the formula $F + V - E = 2$ never fails , over $1700$ years of mathematics could not find an exception.