# Supplement 3.2: Plane and Solid Angles (2/2)

## The Solid Angle

Solid angles come about when the concept of plane angles is extended into the three-dimensional space.
Hence, a plane circle (2D) turns into a sphere (3D); the length of an arc of a circle (2D) turns into
the surface area of a sphere (3D). The solid angle Ω as shown in the illustration below corresponds
to the ratio of the sphere's segment area *a* to the square of the sphere's radius *R*:

Quantities of solid angles are expressed in dimensionless units called steradians with the symbol sr. Quantities in degrees such as in plane angles are not used.

For * small solid angles*, the segment area of a sphere
can be approximated to be considered as a flat surface, thereby simplifying the computation of solid angles.

### Differential solid angles

Just as a plane angle can be expressed as a differential quantity,

so can solid angles likewise be expressed as differentials:

The orientation of a solid angle element in space is best given in
spherical coordinates (or: polar coordinates).
The illustration on the right shows a differential segment area d*a* of a sphere
in the polar coordinates (*r*,*ϑ*,*φ*); by dividing this by the
square of the sphere's radius *R* you get the differential solid angle dΩ.

Given the geometrical relations that apply for the edge of the differential segment area d*a*
assumed as a flat rectangle in polar coordinates (see illustration above), the solid angle can be expressed as:

Many physical processes are mainly dependent on the zenith angle *ϑ*, whereas the azimuth angle
*φ* is irrelevant.
Therefore, the variable *φ* in the equation for d*a* can be eliminated by integration:

The resulting differential segment area d*a* is similar to a belt having the width *R *d*ϑ*
winded around the sphere just below the zenith angle *ϑ* (see illustration below).

The corresponding differential solid angle is: