The pdf for points uniformly distributed on a 3D sphere as a function of the radius r, the polar angle th and azimuthal angle ph is f(r,th,ph) = (3 r^2 sin(th))/(4 pi R^3) (I just normalized over the volume of the sphere and multiplied by the integration measure). Sanity check, int_0^R dr int_0^pi dth int_0^(2 pi) dph f(r,th,ph) = 1 Now, x= r sin(th) cos(ph). By integrating, we find = 0 = (R^2)/5 Therefore, var(x) = (R^2)/5