TITLE: Random point uniform on a sphere
QUESTION [18 upvotes]: If $X=(x,y,z)$ is a random point uniform on the unit sphere in $\mathbb{R}^3$,  Are the  coordinates $x$, $y$, $z$ uniform in interval $(-1,1)$?

REPLY [6 votes]: That this is true can also be seen, by geometry (similar triangles), following the same reasoning that Archimedes (it seems) used to find the surface of a sphere. 


More specifically: if we cut an infinitesimal slice of the sphere, along the coordinate $X$ (vertical in the figure) it's seen that the surface over the sphere equals that of the slice over the surrounding cylinder (this happens only in 3 dimensions, because only in that case the bigger radius for the cilinder compensates the smaller "height" of the slice). Because the random point is uniform over the sphere, then the probability that fall in a interval $[x,x+dx]$ is proportional to the surface, and then you're done.