surface area

The surface area of an n-sphere $\mathbb{S}^n\,$ of radius $r\,$ is

$S_{n} = r^n \frac{2\pi^{\frac{n+1}{2}}} {\Gamma\left(\frac{n+1}{2}\right) }\,$ .

Proof

Following ^[1], in $d\,$ -dimensional spherical coordinates,

$(\sqrt{\pi})^d\,$	$= \left(\int\!dx\,e^{-x^2}\right)^d\,$ ,
	$= \int\!d^dx\,\exp\left( -\sum_{i=1}^d x^2_i \right)\,$ ,
	$= \int\!d\Omega_d \int_0^{\infty}\!dx\,x^{d-1} e^{-x^2}\,$ ,
	$= \left(\int\!d\Omega_d \right)\frac{1}{2}\int_0^{\infty}\!d(x^2)\,(x^2)^{\frac{d}{2}-1} e^{-(x^2)}\,$ ,
	$= \left(\int\!d\Omega_d \right) \frac{1}{2}\Gamma\left(\frac{d}{2}\right)\,$ ,