The "volume" of a hypersphere is given by:

where Γ denotes the gamma function.

The "surface area" is instead given by:

In three dimensions, the formula for the calculation
of volume is:

As for the surface is:

For the definition of

**derivative**, as the limit of the quotient, we have:
the

**surface**is the**derivative**of the**volume**.
And this in each dimension:

*D*(V_{n}) = S_{n-1}
Their relationship in every dimension is remarkably
simple:

V

_{n}/ S_{n-1}= R / n e.g. in 3 dimensions we have: V_{3}/ S_{2}= R / 3
For the calculation of hypervolumes and hypersurfaces,
formulas contain:

**in 2 or 3 dim. appears π, while in 4 or 5 dim. π**

^{2}, 6 or 7 dim. π^{3}**and so on**.

As an example for even number of dimensions, we have:

while for odd number of dimensions: