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.