The volume of an $n$-dimensional ball of radius $R$ has the form

$$V_n(R)=C_nR^n,$$

where $C_n$ is the volume of the unit $n$-ball.

Scaling with radius

If every coordinate is scaled by a factor $R$, $n$-dimensional volume scales by $R^n$. The main problem is therefore determining the constant $C_n$.

Slicing by one coordinate

A ball can be decomposed into lower-dimensional cross-sections. At coordinate $x$, the remaining coordinates occupy an $(n-1)$-ball of radius

$$\sqrt{R^2-x^2}.$$

Thus

$$V_n(R)=\int_{-R}^R V_{n-1}\left(\sqrt{R^2-x^2}\right),dx.$$

This relation turns the geometry of dimension $n$ into an integral involving dimension $n-1$.

A dimension-by-dimension pattern

The first cases are familiar:

$$V_1(R)=2R,$$

$$V_2(R)=\pi R^2,$$

$$V_3(R)=\frac{4\pi}{3}R^3.$$

Continuing the recurrence reveals a regular pattern involving powers of $\pi$ and generalized factorials.

Boundary area

Differentiating volume with respect to radius gives the surface measure of the boundary:

$$S_{n-1}(R)=\frac{d}{dR}V_n(R).$$

The same relationship between accumulated volume and boundary area that holds for ordinary circles and spheres persists in every dimension.