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.

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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!