## Definition of Solid Angle

The solid angle formed by a surface at the origin is the area of the projection of the surface onto onto a sphere of radius 1 centered at the origin.

In the diagram above the solid angle is
$d \omega = \frac{\Omega}{r^2}$

$\mathbf{n}$
is the outward normal at the surface then
$cos \theta = \frac{\mathbf{n} \cdot \mathbf{r}}{r}$

But also
$d \Omega = cos \theta dS = \frac{\mathbf{n} \cdot \mathbf{r}}{r} dS$

Then
$d \omega = \frac{\mathbf{n} \cdot \mathbf{r}}{r^3} dS$

If the origin is not inside
$S$
then
$\omega = \int \int_S \frac{\mathbf{n} \cdot \mathbf{r}}{r^3} dS =0$
and if the origin is inside
$S$
then
$\omega = \int \int_S \frac{\mathbf{n} \cdot \mathbf{r}}{r^3} dS = 4 \pi$
by Gauss's Theorem.