Definition:Solid Angle

Definition

In the words of Euclid:

A solid angle is the inclination constituted by more than two lines which meet one another and are not in the same surface, towards all the lines.
Otherwise: A solid angle is that which is contained by more than two plane angles which are not in the same plane and are constructed to one point.

(The Elements: Book $\text{XI}$: Definition $11$)

Containment of Solid Angle

The three plane angles which together form a solid angle are said to contain that solid angle.

Vertex of Solid Angle

The common vertex of the angles containing a solid angle is known as the vertex of that solid angle.

Subtend

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:

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

where:

$\mathbf {\hat r} = \dfrac {\mathbf r} r$ is the unit vector corresponding to the position vector $\mathbf r$ of the infinitesimal area element $\d \mathbf S$ at $P$

$r$ is the magnitude of $\mathbf r$

$\mathbf {\hat n}$ represents the unit normal to $\d S$.