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A hypersphere is a higher-dimensional analogue of a sphere.

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

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

See also Hypercube 3-sphere