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).

Volume formulas

Consider an (n - 1)-sphere in n-dimensional space with radius r. The n-volume of its interior is

$\frac{2^{\lceil n/2 \rceil}\pi^{\lfloor n/2 \rfloor}r^n}{n!!}$

and the (n-1)-volume of its surface is

$\frac{2^{\lceil n/2 \rceil}\pi^{\lfloor n/2 \rfloor}r^{n-1}}{(n-2)!!}$

where $n!! = n \times (n - 2) \times (n - 4) \times \cdots \times (1 \: or \: 2)$, $\lfloor x \rfloor$ denotes the greatest integer not exceeding x, and $\lceil x \rceil$ denotes the smallest integer not preceding x.

For example, plugging in n = 4 means that a 3-sphere has a 4-volume of $\frac{1}{2}\pi^2r^4$ and a surface volume of $2\pi^2r^3$.

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