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  
\[d \omega = \frac{\Omega}{r^2}\]

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

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

\[d \omega = \frac{\mathbf{n} \cdot \mathbf{r}}{r^3} dS\]

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