A solid angle is two-dimensional by virtue of having no radial position, i.e., the solid angle remains fixed independent of radius. This provides the key to the mathematical formulation for a solid angle. Again, we proceed by scaling up in dimensionality. Recall that a differential angle is defined as follows:

where **dS** is the differential arc length and **r** is the radius.
A differential solid angle is defined by scaling up the differential
arc length to a differential area and dividing by to make the solid
angle independent of radial position (distance from the origin) as follows (see
Figure 7(f)):

Viz., referring to
Figure 7(f),
note that length
**|BC|**
is
generated by sweeping line
(not **r**), through
angle - i.e. the revolution is of a line normal to the vertical
axis
**|AB|**,
about the vertical axis
**|OA|**.
Contrariwise, line
**|BD|**
is generated by sweeping
**|OB| = r**,
through angle
about the origin **O**.