It is well-known that the volume of the unit ball in n-space is $\pi^{n/2}\\!/\Gamma(n/2+1)$. Do you know of a proof which explains this formula? Any proof which does not treat the cases n even and n odd separately (like using an explicit expression for $\Gamma(n/2+1)$ in terms of factorials) should be fine.

It is well-known that the volume of the unit ball in n-space is $\pi^{n/2}/\Gamma(n/2+1)$. Do you know of a proof which explains this formula? Any proof which does not treat the cases $n$ even and $n$ odd separately (like using an explicit expression for $\Gamma(n/2+1)$ in terms of factorials) should be fine.