Hypersphere

A hypersphere of radius r in n-dimensional Euclidean space consists of all points at distance r from a given fixed point (the centre of the hypersphere). This object is an (n-1)-manifold commonly called an (n-1)-sphere. Hence, the special case of an ordinary sphere in three dimensions would be called a "2-sphere".

The "volume" it encloses is

<math>\pi^{n/2}r^n\over\Gamma(n/2+1)</math>

where Γ is the gamma function. The "surface area" of this hypersphere is

<math>2\pi^{n/2}r^{n-1}\over\Gamma(n/2)</math>