In mathematics, a unit sphere is a sphere of unit radius: the set of points at Euclidean distance 1 from some center point in three-dimensional space. More generally, the unit n {\displaystyle n} -sphere is an n {\displaystyle n} -sphere of unit radius in ( n + 1 ) {\displaystyle (n+1)} -dimensional Euclidean space; the unit circle is a special case, the unit 1 {\displaystyle 1} -sphere in the plane. An (open) unit ball is the region inside of a unit sphere, the set of points of distance less than 1 from the center. A unit sphere or unit ball with center at the origin of the space is called the (canonical) unit sphere or the (canonical) unit ball. Any arbitrary sphere can be transformed to the unit sphere by a combination of translation and scaling, so the study of spheres in general can often be reduced to the study of the unit sphere. The unit sphere is often used as a model for spherical geometry because it has constant sectional curvature of 1, which simplifies calculations. In trigonometry, circular arc length on the unit circle is called radians and used for measuring angular distance; in spherical trigonometry surface area on the unit sphere is called steradians and used for measuring solid angle. In more general contexts, a unit sphere is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance", and an (open) unit ball is the region inside.
Unit spheres and balls in Euclidean space In Euclidean space of n {\displaystyle n} dimensions, the ( n − 1 ) {\displaystyle (n-1)} -dimensional unit sphere is the set of all points ( x 1 , … , x n ) {\displaystyle (x_{1},\ldots ,x_{n})} which satisfy the equation
x 1 2 + x 2 2 + ⋯ + x n 2 = 1. {\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}=1.}
The open unit n {\displaystyle n} -ball is the set of all points satisfying the inequality
x 1 2 + x 2 2 + ⋯ + x n 2 < 1 , {\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}<1,}
and closed unit n {\displaystyle n} -ball is the set of all points satisfying the inequality
x 1 2 + x 2 2 + ⋯ + x n 2 ≤ 1. {\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\leq 1.}
Volume and area
The classical equation of a unit sphere is that of the ellipsoid with a radius of 1 and no alterations to the x {\displaystyle x} -, y {\displaystyle y} -, or z {\displaystyle z} - axes:
x 2 + y 2 + z 2 = 1 {\displaystyle x^{2}+y^{2}+z^{2}=1}
The volume of the unit ball in Euclidean n {\displaystyle n} -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. It is
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