You may not know it, but mathematics is all around you in the world today- from the

breakfast you eat in the morning, to the hobbies you enjoy, to the complex world of computers

and games. In this paper, it’s going to be my goal to show you how math is related to the sport

of soccer.

Soccer, in essence, is a fairly simplistic sport. The basic rules are simple, but some of the

more particular ones can become slightly confusing. The MLS (Major League Soccer)

recognizes seventeen basic rules which players and coaches must abide by. However, all of

these are not entirely important to understand the game. First of all, you need a regulation size

ball and two netted goals, eighteen feet by eight feet. Each team consists of eleven players, one

of whom must be the goalkeeper. In addition, both teams are allowed to have a select number of

subs. The number of subs varies depending on the level at which you are playing. No players

are allowed to touch the ball with their hands, besides the goalie, who is only given this privilege

if he is inside the eighteen yard box around the goal. Player uniforms must embody a shirt,

socks, shorts, shin guards, and shoes. In addition the goalkeeper must wear colors that

distinguish him from other players, the referee, and the referee assistants. The game is run by a

main referee and two assistants. The main referee is responsible for control of the game and

his/her decisions regarding facts connected with play are final. The referee’s assistants aid the

referee by indicating offside, when the ball is out of play, and which team gets a corner kick, a

goal kick, or a throw in. Furthermore, they denote when a substitution needs to be made. Game

length for professional games is ninety minutes with two forty-five minute halves. Time is

continuously running. Points (called goals) are awarded to the team that passes the ball

completely over the goal line and into the other team’s goal. Another important part of the game

is the calling of fouls and penalties. “Fouls are called for any of the following six offenses in a

matter that is considered by the referee to be careless, reckless, or using excessive force.

A. Kicks

B. Trips

C. Jumps at

D. Charge

E. Strikes

F. Pushes”(Major)

Anything from free kicks for the opposing team, to game suspensions can be given out for these

offenses. The game is won by the team that has scored the most goals after the entire ninety

minutes of play has expired. If the number of goals scored by each team is the same after the

entire ninety minutes, the game goes into a ten minute overtime with two five minute halves. If

the score remains tied, the game goes into a shoot out until one team has kicked more goals than

the other.

In brief, soccer is played in this way. Now you may look at all of that writing and say

that there is no way that any of this sport could have anything to do with math, but surprisingly

enough, it does.

Math is present in almost anything you do. All sports, games, hobbies, and more have a

number of ways in which they are involved in math. Soccer is not left out. Many mathematical

theories apply to the sport of soccer. To start simple many geometrical shapes are on a soccer

field. The field is rectangular, the goal boxes are rectangular, and the center of the field is a

circle. Even more difficult things can be calculated using math. For instance, the probability of

actually scoring a goal can be calculated by finding the angle to the goal (geometry) and by

finding the center of gravity (physics). More physics applies in calculating the distance and in

what direction a ball will travel when kicked by using projectile motion and initial velocity.

However, in this paper I am going to concentrate on one main focus, and this focus is the shape

of the actual soccer ball itself.

If you actually look at a soccer ball in depth, you will notice that it is an intricate pattern

of pentagons and hexagons covering a spherical surface. This shape is technically called a

truncated icosahedron, a more complex version of a polyhedra.

Basically there are five platonic solids, which are the cube, the tetrahedron, the

dodecahedron, the octahedron, and the icosahedron. “Known to the Greeks, there are only five

solids which can be constructed by choosing a regular convex polygon and having the same

number of them meet at each corner. The cube has three squares at each corner; the tetrahedron

has three equilateral triangles at each corner; the dodecahedron has three equilateral triangles at

each corner; With four equilateral triangles, you get the octahedron, and with five equilateral

triangles, the icosahedron”(Hart).

The number of faces, edges, and vertices can be related to each other using a fairly

simple formula called Euler’s formula. ” The Euler formula reveals a relationship among the

three elements of the polyhedron; vertices, edges, and faces. The Euler formula states that, V,

the number of vertices minus, E, the number of edges plus, F, the number of faces of a

polyhedron is always equal to two”(Koelm). This can be illustrated by looking at the number of

faces, edges, and vertices of the five platonic solids. They are as follows:

faces edges vertices

tetrahedron 4 6 4

cube 6 12 8

octahedron 8 12 6

dodecahedron 12 30 20

icosahedron 20 30 12

To prove the theory correct, take an octahedron for example. If you plug the numbers into

Euler’s formula, you get an equation that looks like this:

6 – 12 + 8 = 2

To firmly establish this theory, use a tetrahedron. When it’s statistics are plugged into Euler’s

formula the equation looks like this:

4 – 6 + 4 = 2

Again the equation is equal to two. If the math is done correctly, those answers should be the

correct ones. This formula also works for the soccer ball, namely the truncated icosahedron.

After carefully counting all the vertices, faces, and edges, I found that the truncated icosahedron

has these characteristics:

faces edges vertices

truncated icosahedron 32 90 60

The soccer ball falls into the category of a archimedean semi-regular polyhedra “A key

characteristic of the Archimedean solids is that each face is a regular polygon, and around every

vertex, the same polygons appear in the same sequence”(Hart). To more fully understand the

soccer ball is to begin with the regular icosahedron itself, which is one of the five platonic solids

listed earlier. This shape by definition has twenty faces, each being an equilateral triangle. The

faces are arranged so that five triangles meet at a vertex at the top and bottom. There are five

such vertices throughout this particular shape. The truncated part of the soccer ball can be

explained best by the illustration that follows, but basically it is defined like this: Each vertex is

cut off along a plane perpendicular to the radius at the vertex. Since the vertex is formed by the

intersection of five triangles, the new facets created by this cutting are pentagons. Then the

remaining portions of the triangle are converted into hexagons.

Now if you put that all together, you have to wonder how flat polygons can be put

together to form a spherical ball. The answer to this question is fairly simple. The angles of the

sides of a hexagon are 120 degrees, while the measures of the angles of the sides of a polygon

are 108 degrees. Since the ratio of hexagons to polygons on a soccer ball is roughly 2:1, you

would add up the angles in this way:

120 + 120 + 108 = 348

If you’ll notice the angles add up to be 348 degrees. 360 degrees is the measure of a flat surface.

Because the angle measurement of the addition of the three angles equals 348, a number less

than 360, which is that of a flat surface, the result when all thirty faces are fit together in this

form is a spherical shape.

Overall, that is basically the way that a truncated icosahedron is created. Basically, it

starts off as a regular polyhedron, and then is cut at it’s vertices until the shape of the soccer ball

is created.

After reading this paper, I hope that you’ll realize the significance of mathematics in

sports. Most people, when told that there is a lot of math involved in daily activities, laugh at the

whole concept of it. Not just soccer, but numerous things can be applied to mathematics. This

paper hopefully was a good example of just how much soccer was related to math. Furthermore,

this was only one small aspect of the mathematical side of soccer. Subsequent theories and ideas

could be applied to many other aspects of the game. So just remember, math is not just some

pointless classes you take in school, it can be applied ANYWHERE!