# Averaging over a solid angle

How does one averages a function over a solid angle? I'm used to do this with conventional angles but it's the first time I come across this problem with a solid angle.

In my specific problem, the function is only dependent on the azimuthal angle, which should simplify the problem somewhat and the average is over $4\pi$ steradians, but a generic solution would be greatly appreciated.

Similarly to a 2D angle, you integrate the function over the solid angle of interest and divide by the solid angle. If it depended upon both $\theta$ and $\phi$ and you were interested in the whole sphere, you would do $\overline{f}=\frac{\int f(\theta,\phi)\cos(\theta)d\theta d\phi}{4\pi}$. If your function does not depend upon $\theta$, that integral is 2 ($\cos\theta$ from $0$ to $\pi$) and now you have a usual 2D style average. So $\overline{f}=\frac{\int f(\phi) d\phi}{2\pi}$