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 $$ \Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}\ ? $$ (I'll post my own answer, but I know there are many ways to show this, so post your own!)