Solid Angle Definition


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What is Solid Angle?

Solid angle definition. The solid angle, $\Omega$, subtended by a complete sphere at the center is $4\pi$. The solid angle subtended by an arbitrary area at a point is $4\pi$ times the fraction that such an area is of the complete area of a sphere centered on that point.

\[
{\rm d}\Omega = \sin{\theta}{\rm d}\theta{\rm d}\phi, \ \ \ \Omega = \int_{S}{\sin{\theta}{\rm d}\theta{\rm d}\phi}
\]

Another way of putting it is that if we have an area ${\rm d}S$, say, that is at a distance $r$ from a reference point, $P$, then since ${\rm d}S/(4\pi r^{2})$ is the fraction of the area of the complete sphere centered on $P$,
\[
{\rm d}\Omega \ = \ 4\pi \left( \frac{{\rm d}S} {4\pi r^{2}} \right) \ = \ \frac{{\rm d}S}{r^{2}}
\]