A hypersphere is the set of all points equidistant from a single point in a certain n-dimensional space. For a particular value of n, we have an (n - 1)-sphere (because although the figure itself exists in an n-dimensional space, the hypersphere is a 'surface' in n-dimensions and therefore an (n - 1)-dimensional object).
Consider an (n - 1)-sphere in n-dimensional space with radius r. The n-volume of its interior is
and the (n-1)-volume of its surface is
where , denotes the greatest integer not exceeding x, and denotes the smallest integer not preceding x.
For example, plugging in n = 4 means that a 3-sphere has a 4-volume of and a surface volume of .