The ordinary angle, measured in radians, is the length of the arc of a
unit circle subtended by the angle. A full circle, 360 degrees,
is 2 π radians, of course  the length of the perimeter of a unit
circle.
Solid angles are measured in "steradians";
instead of the arc length of the portion of the unit circle subtended
by the angle, it's the area of the unit sphere subtended by the solid
angle.
The solid angle subtended by a
cone whose apex has
angle 2θ is obtained by making the cone 1 unit high, setting it inside
the unit sphere, and finding the area of the part of the sphere the
cone is sitting on. More simply, to find the area of a part of a
sphere of radius R subtended by a particular (linear) angle, we just
integrate (see
figure 1 below):
(1)
The solid angle subtended by the cone with apex 2θ is the area of the
unit sphere subtended by the cone, so we just set R=1, to obtain
(
2)
For
theta = π, which would include the entire sphere, (2) evaluates to
4π  and so we see there are 4π steradians in a full sphere.
For a
small angle, we can use the second order approximation to cos(θ)
(3)
and then the solid angle subtended by the cone is approximately
(
4)
This
comes up in optics, where we want to know what fraction of the light of
a point source is received by a pupil which has angular radius θ when
viewed from the location of the point source. In general, it's
the solid angle subtended by the pupil, divided by the solid angle of
the entire sphere. That's
(5)
or, for small angles, it's approximately
(6)
Figure 1: Integrating over part of a sphere

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created on 9/22/2007