# Definition:Solid Angle

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## 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:

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

where:

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