Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$? It seems as if no one has asked this here before, unless I don't know how to search.
The Gamma function is
$$
\Gamma(\alpha)=\int_0^\infty x^{\alpha-1} e^{-x}\,dx.
$$
Why is
$$
\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}\text{ ?}
$$
(I'll post my own answer, but I know there are many ways to show this, so post your own!)

This is a "proof". We know that the surface area of the $n-1$ dimensional unit sphere is 
$$
|S^{n-1}| = \frac{2\pi^{\frac{n}2}}{\Gamma(\frac{n}2)}.
$$
On the other hand, we know that $|S^2|=4\pi$, which gives
$$
4\pi = \frac{2\pi^{\frac32}}{\Gamma(\frac32)} = \frac{2\pi^{\frac32}}{\frac12\Gamma(\frac12)}.
$$
