Solid Angle

The solid angle subtended by a surface is defined as the surface area of a unit sphere covered by the surface's projection onto the sphere. This can be written as

(1)

where is a unit vector from the origin, is the differential area of a surface patch, and is the distance from the origin to the patch. Written in spherical coordinates with the colatitude (polar angle) and for the longitude (azimuth), this becomes

(2)

Solid angle is measured in steradians, and the solid angle corresponding to all of space being subtended is steradians.

To see how the solid angle of simple geometric shapes can be computed explicitly, consider the solid angle subtended by one face of a cube of side length centered at the origin. Since the cube is symmetric and has six sides, one side obviously subtends steradians. To compute this explicitly, rewrite (1) in Cartesian coordinates using

			(3)
			(4)

and

			(5)
			(6)

Considering the top face of the cube, which is located at and has sides parallel the - and -axes,

			(7)
			(8)

as expected.

Similarly, consider a regular tetrahedron with side length , the origin at its geometric centroid, and base at , where is its inradius. Its bottom vertices are at and , where

			(9)
			(10)
			(11)

Then runs from to , and for the half of the base in the positive half-plane, can be taken to run from 0 to , giving

			(12)
			(13)

i.e., , as expected.