# Calculate Pentagons and Hexagons in a Soccer Ball

There are many pentagons and hexagons in a soccer ball. Now, let’s calculate them.

## Euler’s formula

On polyhedron, the following equation is valid.

(vertex count) – (edge count) + (face count) = 2

The proof of Euler’s formula is written on Think about Euler’s Formula on Polyhedron.

Now, using this formula, calculate the number of pentagons and hexagons. ( I calculated faces, etc. without Euler’s Equation on Soccer Ball and Regular Polyhedron . )

## Calculate

Define $$m$$ as the number of pentagons, and $$n$$ as one of hexagons.

On soccer ball, one vertex is common in 3 planes. So, the number of vertexes on a soccer ball is expressed as $$\frac{5m + 6n}{3}$$, the number of edges is $$\frac{5m + 6n}{2}$$, the number of faces is $$m + n$$ .

According to Euler’s formula, the following equation is valid.

$$\frac{5m + 6n}{3} – \frac{5m + 6n}{2} + m + n = 2$$

Now, calculate the above equation and get the value of $$m$$ .

$$m = 12 .$$

On a soccer ball, one pentagon neighbors 5 hexagons, and one hexagons neighbors 3 pentagons. Then, $$n$$ is calculated as the following.

$$n = \frac{5 m}{3} = \frac{5 \times 12}{3} = 20 .$$

From the above, the number of hexagons is 10, and the number of pentagons is 20.

You can calculate them with Descartes theorem. I wrote it on Proof of Descartes Theorem.

## Note

The shape of Fullerene, C60, is the same as soccer ball. Let’s check whether the number of vertexes is 60.

$$\frac{5m + 6n}{3} = \frac{5 \times 12 + 6 \times 20}{3} = 60 .$$

## Reference

Euler’s formula is explained in the following book.