Variance of the height of a random point on a sphere

A point $(x,y,z)$ is chosen uniformly at random on the surface of the unit sphere. What is the variance of its $z$-coordinate?

$1/3$. By symmetry $\mathbb{E}z=0$ and $\mathbb{E}x^2=\mathbb{E}y^2=\mathbb{E}z^2$; since $x^2+y^2+z^2=1$, each equals $\tfrac13$. (Archimedes' theorem also says $z$ is uniform on $[-1,1]$, whose variance is $\tfrac13$.)