I wrote a longer essay here, but a dear reader told me it was too much, so here's the short version.
Steradians are the measure of a solid angle. 12.5663 steradians is the inner solid angle of a whole sphere. (This is calculated from 4 times pi.) This is analogous to 6.2832 radians (2 times pi) being the inner angle of a whole circle.
Do any of you math whizzes have the solution to this: What is the plane angle across a cone that includes exactly 1 steradian of solid angle? I'm guessing it's about 70 to 80 degrees.