How is math used in Soccer?

Mar 23, 2022 | Location Pacific Heights San Francisco

While many viewers may not be doing extensive calculations while watching a game, math is heavily involved in the game of soccer!. In the aftermath of a game, whether it was a dull draw with few highlights or a spectacular 7-1 result in a world cup semi-final, not many would bring out their old statistics books in order to try and explain what happened.  

Football analytics may not be as exciting as actually playing a game! But it can be just as fun to model successful strategies using statistics and probabilities while watching your favorite teams play. In addition to being entertaining, learning about these techniques allows you to put your analytical abilities to the test when deciding on which teams/players should win in any given matchup. And who knows - maybe you'll even want to try and apply some of these analytical skills when creating a strategy for betting on games in the future!


However, like everything else on this planet, soccer can be viewed through the eyes of mathematics. There is math involved in ranking players, winning penalties, optimal angle for a throw-in, score system, shape and dimensions of pitch, and leagues. Today we will go over a few of the mathematical processes that make the game what it is..


The perfect ball


The familiar black and white checkered soccer ball are made from many leather pieces- 12 black pentagons and 20 white hexagons, all of which are regular. On a standard soccer ball, the 2D shapes are tessellated together to leave no gaps between them. The pentagons and the hexagons must have an equivalent length of sides to create this visual effect. In addition to this, for all shapes to tessellate the angles on the corners must add up to 360 degrees!


Tessellation, 2D shapes, and angles have a massive role in creating the standard soccer ball.


Optimal angle for a throw-in


The angle which will maximize the distance of a throw-in is around 30 degrees. In mechanics, people learn that 45 degrees are the usual angle that will maximize the distance of a projectile regardless of speed. Soccer players can usually throw at higher speeds from lower angles meaning that the best angle is around 30 degrees, contrary to the standard 45 degrees.


Spain’s tiki-taka


The tiki-taka is an offensive maneuver and a great example of using geometry in real-time to create space and beat defenders. It is a systems approach to soccer, based on team cohesion and an understanding of the area of a soccer field. The soccer players try to form triangles all around the pitch to maintain the ball possession, making it difficult for the opponent to obtain the ball and organize their game. Tiki-taka has proven to be very successful as a soccer strategy. Barcelona won the sextuple in 2009 by using the Tiki-taka style. They played with a high defensive line usually applying the offside trap with midfielders providing support to defenders.


Eyes on the prize. Goal keeper's one on one


Distracting a striker provides an excellent example of soccer and math working together. The goal is to make the space available for scoring smaller, which lowers the probability of a striker making a goal. The goalie often charges toward a striker in a one-on-one situation to close the space, reducing the angle and space available to strike the ball. This is another concept of mathematical soccer. Goalies are typically the last line of defense to keep the opponent from scoring.


Beating the goalie’s charge


Chipping a charging goalie is one of the most beautiful moves in soccer. As space is reduced, the striker sees the possibility of scoring. A 3-dimensional view allows him to kick over the goalie's head, and into the net. There's nothing difficult about a chip shot as requires a deft touch that follows a perfect parabola into the net.


Saving penalties


As we’ve explained, Soccer is a treasure trove of mathematical concepts, but did you know that statistics are the key to successful penalty kick defenses? The teams study one another penalty shots to develop an understanding of their shot pattern. Once a goaltender understands a particular player's pattern, they can predict where they are going to place a shot to block it.


The diagonal system of control


Professional soccer referees use a system called the diagonal system of control (DSC), in which two assistant referees move up and down diagonally opposite touchlines while the referee moves in a diagonal direction from Southeast to Northwest. According to geometry, this formation covers more of the pitch than if the referee just ran up and down the center of the pitch. For instance, if the referee merely ran up and down the middle of the pitch, there would be parts of the pitch that are not covered.


  • How do we know that soccer referees are happy?

  • Because they whistle while they work.


Consider the math involved in every tick of the scoreboard clock the next time you watch a soccer game. The clock is full of math itself, but that will have to be for another article.

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