TITLE: The geometrical meaning of the common value in the law of sines in hyperbolic geometry
QUESTION [5 upvotes]: What is the geometrical meaning of the common value in the law of sines, $\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c}$ in hyperbolic geometry? I know the meaning of this value only in Euclidean and spherical geometry. 
EDIT, Will Jagy. The OP is looking for some fourth fairly natural real number that can be calculated from a triangle, that gives the same answer as the common value in the Law of Sines. The original question is at https://math.stackexchange.com/questions/69345/the-law-of-sines-in-hyperbolic-geometry

REPLY [3 votes]: There is a meaning, though whether it is geometric is up to you to decide. You can see it in:
http://mathworld.wolfram.com/GeneralizedLawofSines.html
which is fairly incomprehensible without
http://mathworld.wolfram.com/HyperbolicPolarSine.html
To see how you might derive such a thing, see this:
http://arxiv.org/abs/math/0211261