(Recurrent) volume of a sphere

[Tolaso J Kos]

Let \( V_n(R) \) be the volume of the sphere of center \( 0 \) and radius \( R \) in \( \mathbb{R}^n \). Prove that for \( n \geq 3 \) holds:

$$V_n(1)= \frac{2\pi}{n} V_{n-2} (1)$$

[Hint: Apply Cavaliere's principal twice and after you prove \( V_n (R)= R^n V_n(1) \) convert to polar coordinates ]

[Papapetros Vaggelis]

By definition holds:

$$ V_n(1)=\idotsint\limits_{x_1^2+x_2^2+\cdots+x_n^2=1} 1\, {\rm d}(x_1, \; x_2, \dots\, x_n)$$


Converting into spherical coordinates and through the Jacobian we have that:

$$V_{n}(1)=\int_{0}^{2\,\pi}\,\int_{0}^{\pi}...\int_{0}^{\pi}\,\int_{0}^{1}r^{n-1}\sin^{n-2}\theta_1\,\sin^{n-3}\theta_{2}...\sin\theta_{n-2}\mathrm{d}r\mathrm{d}\theta_1...\mathrm{d}\theta_{n-1}$$


The final expression is an expression containing single integrals which are reduced down, due to Walli's formula, in a Gamma form. Indeed:

\(\displaystyle{V_{n}(1)=\dfrac{\pi^{n/2}}{\Gamma\,\left(1+\dfrac{n}{2}\right)}\,\,,n\in\mathbb{N}}\) .

\(\displaystyle{V_{1}(1)=\dfrac{\sqrt{\pi}}{\Gamma\,\left(1+\dfrac{1}{2}\right))}=\dfrac{2\,\sqrt{\pi}}{\Gamma\,(\dfrac{1}{2})}=2}\)

since \(\displaystyle{V_{1}(1)=\mu\,\left(\left\{-1,1\right\}\right)=0}\) .

\(\displaystyle{V_{2}(1)=\dfrac{\pi}{\Gamma(1)}=\pi}\) ( circle or radius \(\displaystyle{1}\)) .

If \(\displaystyle{n\geq 3}\), then :

$$\begin{aligned} V_{n-2}(1)&=\dfrac{\pi^{(n-2)/2}}{\Gamma\,\left(1+\dfrac{n-2}{2}\right)}\\&=\dfrac{\pi^{n/2}\cdot \pi^{-1}}{\Gamma\,\left(\dfrac{n}{2}\right)}\\&=\dfrac{n}{2}\,\dfrac{1}{\pi}\,\dfrac{\pi^{n/2}}{\displaystyle{\dfrac{n}{2}\,\Gamma\,(\dfrac{n}{2})}}\\&=\dfrac{n}{\,2\,\pi}\,\dfrac{\pi^{n/2}}{\Gamma\,\left(1+\dfrac{n}{2}\right)}\\&=\dfrac{n}{2\,\pi}\,V_{n}(1)\end{aligned}$$

so: \(\displaystyle{V_{n}(1)=\dfrac{2\,\pi}{n}\,V_{n-2}(1)\,\,,n\geq 3}\) .

In more general it holds that:


$$V_{n}(R)=\dfrac{2\,\pi\,R^{n}}{n}\,V_{n-2}(R)\,\,,n\geq 3$$

and \(\displaystyle{V_{1}(R)=2\,R\,\,,V_{2}(R)=\pi\,R^2}\) .


(due to Walli's formula)