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Let be the surface area of the unit -sphere of radius embedded in -dimensional Euclidean space, and let be the volume of its interior, the unit -ball. The surface area of an arbitrary -sphere is proportional to the st power of the radius, and the volume of an arbitrary -ball is ...
where S n − 1 (r) is an (n − 1)-sphere of radius r (being the surface of an n-ball of radius r) and dA is the area element (equivalently, the (n − 1)-dimensional volume element). The surface area of the sphere satisfies a proportionality equation similar to the one for the volume of a ball: If A n − 1 ( r ) is the surface area of an ( n ...
The volume of the unit ball in Euclidean -space, and the surface area of the unit sphere, appear in many important formulas of analysis. The volume of the unit n {\displaystyle n} -ball, which we denote V n , {\displaystyle V_{n},} can be expressed by making use of the gamma function .
A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n−1)-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a ...
The rectangular field of play used for American football games measures 100 yards (91.44 m) long between the goal lines, and 160 feet (48.8 m) (53.3 yards) wide. The field may be made of grass or artificial turf. In addition, there are two end zones on each end of the field, extending another 10 yards (9.144 m) past the goal lines to the "end ...
All competitions in the halfcourt game of 3x3, whether men's, women's, or mixed-sex, use a dedicated ball with the circumference of the women's ball but the weight of the men's ball. Rock – the ball; Basketball court – the playing surface, consisting of a rectangular floor with baskets at either end. In professional or organized basketball ...
A typical basketball. A basketball is a spherical ball used in basketball games. Basketballs usually range in size from very small promotional items that are only a few inches (some centimeters) in diameter to extra large balls nearly 2 feet (60 cm) in diameter used in training exercises.
A football pitch or soccer field is the playing surface for the game of association football. Its dimensions and markings are defined by Law 1 of the Laws of the Game, "The Field of Play". [1] The pitch is typically made of natural turf or artificial turf, although amateur and recreational teams often play on dirt fields. Artificial surfaces ...