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

See also Hypercube 3-sphere