A solid angle
(measured in steradians "sr") is assigned to the cone generated by half a straight line
originating at the center of a sphere of unit radius with one point of that line moving
in a closed loop at the surface of the unit sphere.
The measure of such a solid angle is simply the spherical surface area enclosed by
the aforementioned loop at the surface of the unit sphere.
Just like a planar angle, a solid angle can be oriented
(i.e., assigned a sign) according to the direction in which the loop is traveled.
The usual convention is to count a solid angle positively if the loop is traveled
clounterclockwise when seen from the outside of the sphere or, equivalently,
clockwise seen from the origin at the center of the sphere.
You may memorize this by recalling that the south face of a loop
is seen at a positive solid angle (using the usual
convention to define the "north" and "south" side of an oriented loop).
That loop may cross itself many times: The spherical area so
enclosed is tallied algebraically as in the planar case we
describe elsewhere.
Just like planar angles are defined modulo
2p,
solid angles are defined up to a multiple of 4p,
because such is the entire surface area of a unit sphere
A "spat" is the solid angle subtended by the whole sphere
(4p).
Indeed, consider that a solid angle A changes to -A
when you reverse the direction of its defining loop.
However, you could also consider that the solid angle has become
(4p-A)
because the new orientation of the loop makes it enclose (as its "south side")
whatever part of the sphere was not previously enclosed by the loop as originally
oriented.
You may use this argument to convince yourself that multiples of
4p
are as irrelevant to solid angles as multiples of
2p
are irrelevant to planar angles.
The steradian is not the only unit of solid angle.
Astronomers routinely express solid angles in square degrees,
they also use square minutes or square seconds
for tiny solid angles.
Indeed, if a "rectangular" patch of sky is so small that the curvature of the celestial
sphere is negligible, then its surface is almost flat and it has an area very nearly equal
to the product of its angular width by its angular height (technically,
those concepts of "width" and "height" become precise only in the context of that flat
approximation). The result is in steradians if those angles are given
in radians. On the other hand,
if such angles are given in degrees, then the result is, by
definition, obtained in "square degrees". The square degree
is thus just a practical unit of solid angle which could be used to measure solid
angles of any size, although the aforementioned "small angle"
computation is only valid for very tiny rectangular patches of the sphere.
1 square degree =
( p / 180 ) 2 =
0.0003046... sr
A square minute is 602 = 3600 times smaller
than that.
A square second is 12960000 times smaller than a
square degree; it's roughly 2.35 10 -11 sr.
The whole celestial sphere (twice the sky) corresponds to a solid angle of
1 spat = 4p [sr] =
41252.96... square degrees
