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Geometry in the game

This content is provided by the 49ers Museum Education Program.


Math can be seen everywhere. Math is present in our world all the time --in school, in the workplace, in our homes -- and in the sports we play. The study of mathematics has been developed from a combination of thoughts and ideas from a variety of scholars throughout the ages and from all over the world. Math helps us in so many ways, such as assisting us in understanding patterns, improving communication and showing how interconnected the world really is. We study and practice math so that we can solve problems and create new technologies to improve our society.
Without understanding math, it is impossible to understand the game of football: people and objects move in lines and angles, times are recorded, points are counted, and statistics are calculated. Football players should be interested in the mathematics of the game because it can help them win, and math can help fans better understand how the game is played. Another example of how math influences the game is its relationship to engineering.
The study of geometry can explain the reasoning behind how certain designs of architecture (such as in the design of football stadiums), have been developed and used to support fan viewing. Let’s explore how geometry is directly tied to football, but first, let’s understand what geometry really is.
Geometry is mathematical study that deals with shapes, sizes, angles, positions, and properties of space. Geometric shapes, like circles and squares, are forms often seen in math. You can easily spot them, even if they move, rotate, or get bigger or smaller. Geometric shapes are very much a part of our environment and help explain the ways in which we view and understand our world. Geometry is one of the oldest mathematical sciences and is used in constructing structures like buildings, cars, ships; geometry was especially utilized in the engineering and design of Levi’s® Stadium.

Geometry in stadium design

Most stadiums, and a lot of other structures, are actually built with geometric four-sided shapes (like rectangles and squares) because they are easy to work with; however, to avoid excessive weight on the structure itself, many engineers utilize the shape of a triangle in their design. The use of triangles in a construction project (like a stadium) makes the structure much stronger than any other shape that can be used because it can effectively support both vertical and horizontal loads (weight).
The fact is, triangles are easier to join together than other shapes, and engineers have found that the fewer amount of sides a shape has, the sturdier it will be. If a square is used to try to support a weight, it will likely have its top section collapse, while a triangle can distribute the force of the load down its sides.
Triangular structures in most cases cost less, and particularly for stadiums, weigh less than a structure built with other shapes that utilize more angles. The more angles a shape has, the more it will weigh. The use of triangles and other geometric forms were a vital part of the design and overall construction of Levi’s® Stadium to support up to 70,000 fans.
Engineers use geometric shapes like triangles in the design and construction of Levi’s® Stadium. Sports stadiums come in a variety of shapes and patterns, with many designs capitalize on the fundamental strength that triangular shapes provide for its support.
Geometric figures are common in the world in which we live. Various sports have a field shape and markings that are specific to the sport but involve a multitude of geometric forms. Each shape serves its own purpose and has specific meaning. For instance, the arc on a basketball court shows players which shots are worth two points and which ones are worth three. The circles in the center of a soccer field shows players where to line up to start the game, and in football, the parallel line formations determine yardage.
Football uses items of both two- and three-dimensional shapes, from the equipment used to the playing fields on which the game takes place. Whether they realize it or not, football players use geometric dimensions of the field. The rectangular football field is divided into three smaller rectangles by two goal lines at either end of the field. In addition to knowing the shapes of the field there are also specific parameters. The diameter of the field goal posts (18 feet, 6 inches wide), the circumference of the football (22 inches), and the width of the field (53 yards), are all standard measures that must be followed in any football game. The playing field also places heavy emphasis on angles, by putting players into the proper positions to move the ball and score points.
Geometrical shapes are quite common in football. The playing field for the game of football is made up of rectangles and is divided by parallel lines marking 10 yard increments. Also, the throwing, catching and kicking of the geometric form called a prolate spheroid (the football) defines the skill that is needed to play the game.
An angle is formed when two lines meet (or intersect) at a point. If you look around, angles can be seen all around you, in the products we use and in the buildings we reside in. All designs, whether they are for the construction of a building or in the development of new products, go through the process of measuring from one point to another, to ensure that every design component fits together accordingly. There are a various classifications of angles, and each are classified according to their size. When we work with angles, we identify them as a measurement of degrees.
A degree is just a fraction of a circle. A circle has 360 degrees and a half circle equals 180 degrees. The amount of turn (or degrees) from one arm of the angle to the other determines the size and classification of an angle. The most common types of angles are acute, right and obtuse angles.
A right angle equals 90 degrees exactly. An acute angle is greater than 0º and less than 90º. Obtuse angles are angles greater than 90 degrees. Angles are measured by a math tool called a protractor. A protractor is a measuring instrument that has the shape of a semi-circle and is marked with two scales marked from 0º to 180º to measure all types of angles.
After learning about the different types of angles, can you classify the above angle examples?
Angles play a crucial role in football, from a pass or kick, to the way players move and how plays are called. Football players can even use angles to enhance their individual or team performance. For example, a football placekicker must know what angle to kick the ball at to score a field goal. The path the football will take once it’s kicked comes down to the angle at which it is kicked, the amount of force applied, and the height of the kick. When kicking from thirty yards away, a larger angle is necessary to get the ball through the goalpost. However, when making a fifty yard kick, a shorter angle is called for, so the ball can travel the greater distance to the goal. When the opposing team is trying to block the kick, a higher angled kick is necessary.
By positioning their bodies strategically using angles, players can evade defensive tackles.
A quarterback must also know the approximate angle to make a successful pass to a receiver. The angle at which the football is thrown is determined by the angle made by the extension of the quarterback's arms to the rest of his body.
Understanding angles is also important for good offense and defense. Basically, the offense tries to create angles, and the defense tries to cut down angles. When a defensive player runs onto the field, he must calculate the angle to determine where to best tackle or pursue an offensive player. The way a player can evade the opposing team’s defense can determine the outcome of the game.
Angles help predict how the players move, and also determine how to defend a player. Facing the player directly will give the player greater space to move on either side. However, facing the player at an angle will reduce their ability to move in certain directions. Professional football players instinctively apply the concept of angles to achieve the most successful outcomes in the game. If you think about it, angles are an important aspect of any sport.
The game of football provides numerous opportunities to see geometry at work. No matter what the form, geometry helps us understand specific experiences and it especially helps us define and enjoy the games we play.
This content is provided by the 49ers Museum Education Program.

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