Mathematics of volleyball

Summary: Mathematics is fundamental to player motion, strategy, and scoring in volleyball.

Volleyball, which began in the late nineteenth century as a non-contact recreational sport, quickly developed into a globally popular competitive sport. Two teams, typically with two to six players, face one another on opposite sides of a rectangular court divided by a net. Beach volleyball is played on sand courts rather than a hard surface. Game strategy uses mathematical concepts such as angles, rotation, and parabolic motion in an effort to impart optimal trajectories, speeds, and spins on the ball to prevent the other team from successfully returning it. The receiving team must understand three-dimensional motion and vectors in order to intercept the ball and change its direction, often using a sequence of hits coordinated among several players. The strategies of beach volleyballers often differ from those of hard court volleyballers because of differences in the ability to jump or dive for an incoming ball. Mathematics is also used to analyze and model body kinetics, such as the motions of a player’s shoulders and arms while serving. Statistics are used to analyze and describe both team and individual proficiencies and success. These include measures like number of attacks, kills, and assists; hitting percentages; and kill average and efficiency as a function of total attempts.

General Game Play and Scoring

Volleyball teams work together to hit the ball over the net in such a way as to prevent the other team from returning it. A match consists of three or five games. The third game of a three-game match or the fifth game of a five-game match is the deciding game. A single sequence of back and forth hitting is known as a “rally,” which begins with one side serving the ball and ends when one team or the other fails to legally return it. Each side gets three attempts and the same player may not touch the ball twice in a row. At the end of the rally, the winning side may earn a point, the right to serve the ball, or both.

There are two different scoring systems used in volleyball. In side-out scoring, only the serving team may earn a point. In rally scoring, either side earns a point. Winners always get the serve. Deciding games are played to 15 points; nondeciding games are played to 25. However, the winning team must be ahead by at least two points or play continues. Sometimes, a scoring cap is used, which nullifies this requirement. Statistical analyses show that rally point scoring makes matches shorter and match lengths more predictable versus side-out scoring. However, there appears to be no significant effect on scoring margins between teams; on average, after an even number of serve changes, points awarded to non-serving teams balance. In addition to statistics, Markov chains are useful for analyzing volleyball games in terms of the proportion of points won and the probabilities of winning a point, game, and match.

Player Roles and Strategy

Hard court teams typically consist of six players with specialized roles, with the left, center, and right forwards in a row along the frontcourt and the left, center, and right backs in a row along the backcourt. However, players usually rotate through positions during play, requiring analysis of permutations and the timing of substitutions. Beach volleyball teams typically consist of two players each, generally front and back. Players seek to control the ball through the angle, force, and timing with which the ball is struck and by choosing whether or not to impart spin on the ball. The volleyball typically travels along a parabolic path, modified by its spin and additionally influenced by player efforts and external factors, such as air resistance. The basic skills used in volleyball include the serve, pass, set, spike, block, and dig. A variety of serves can be used as the server hits the ball into the opponent’s court. Different types of serves affect the ball’s direction, speed, and acceleration with the goal of increasing the difficulty of handling the ball for the opposing team. Serves that have flatter parabolic paths tend to preserve more of the initial force and velocity and are usually more difficult to return.

The opposing team’s first reception of the ball is known as the “pass,” the second contact is known as the “set,” and the third contact is known as the “attack” (also called “spike”), though a team may not opt to use all three contacts in every play. A block is a team’s attempt to prevent the opposite team from spiking the ball into their court, and a dig is an attempt to prevent a ball from hitting the court. Shots include the hard angle, deep angle, seam shot, line shot, angled line shot, swiping shot, high and hard, and the save. Achieving different shots relies on affecting the ball’s speed, spin, and angle of trajectory through shoulder and hip positions, aiming at gaps between opposing players, and the amount of force applied. Spin tends to make the ball more difficult to return successfully, since the appropriate counterforce to control the ball and change its directional vector is more difficult to determine and apply quickly.

Bibliography

Calhoun, William, G. R. Dargahi-Noubary, and Yixun Shi. “Volleyball Scoring Systems.” Mathematics and Computer Education (Winter 2002).

Kiernan, Denise. Sports Math. New York: Scholastic Professional, 1999.

USA Volleyball. Volleyball Systems and Strategies. Champaign, IL: Human Kinetics, 2009.