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
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
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 symmetrical and has six sides, one side obviously subtends 
steradians. To compute this explicitly, rewrite (1) in Cartesian coordinates using
and
Considering the top face of the cube, which is located at 
and has sides parallel the 
- and 
-axes,
as expected.
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
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