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
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 symmetrical and has six sides, one side obviously subtends steradians. To compute this explicitly, rewrite (1) in Cartesian coordinates using
Considering the top face of the cube, which is located at and has sides parallel the - and -axes,
Similarly, consider a tetrahedron with side lengths with origin at the centroid, base at (where is the centroid), and bottom vertices at and , where
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
i.e., , as expected.
NOTE: obtained from http://mathworld.wolfram.com/SolidAngle.html