= Solid angle
{title2=$\Omega$}
{wiki}
A <solid angle> measures how much of the surrounding directions a region occupies, as seen from a specified point $O$. Draw a <sphere> of <radius> $R$ centered at $O$ and mark where those directions meet its surface. If the marked patch has <surface area> $A_R$, then
$$
\boxed{\Omega=\frac{A_R}{R^2}.}
$$
A <Euclidean similarity> scales both $A_R$ and $R^2$ by the same factor, so the chosen <radius> does not affect the answer. On the unit <two-sphere>, the numerical value is simply the patch's <surface area>. Unlike an <angle> drawn in one plane, a <solid angle> measures coverage in two angular directions.
For example, the full <sphere> has <surface area> $4\pi R^2$ and hence <solid angle> $4\pi$ <steradians>. A <hemisphere> occupies $2\pi$ <steradians>. Three mutually perpendicular coordinate planes divide the directions into eight congruent pieces, so each octant occupies $4\pi/8=\pi/2$ <steradians>. A <solid angle> of one <steradian> therefore covers $1/(4\pi)\approx7.96\%$ of all directions, rather than one full turn.
This makes a geometric question answer a collection question. For an ideal <point emitter> with <isotropy>, equal <solid angles> carry equal <radiant flux>. A detector accepting directions of total <solid angle> $\Omega$ intercepts the fraction $\Omega/(4\pi)$ of that emitted <radiant flux>, before losses. With direction-dependent <radiant intensity> $I$, the accepted <radiant flux> is instead $\int I\,d\Omega$. The <solid angle of a disk> below gives an exact example where the familiar <inverse-square law> is only a distant-source approximation.
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