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Plane Angle

Let $AB$ be a line segment and $C$ be a point:


The line segment $AB$ is said to subtend the angle $\angle ACB$.

Solid Angle

Let $S$ be a surface oriented in space.

Let $P$ be a point in that space.

The solid angle subtended by $S$ at $P$ is equal to the surface integral:

$\ds \Omega = \iint_S \frac {\mathbf {\hat r} \cdot \mathbf {\hat n} \rd S} {r^2}$


$\mathbf {\hat r} = \dfrac {\mathbf r} r$ is the unit vector corresponding to the position vector $\mathbf r$ of the infinitesimal surface $\d S$ with respect to $P$
$r$ is the magnitude of $\mathbf r$
$\mathbf {\hat n}$ represents the unit normal to $\d S$.