The law of sines in hyperbolic geometry What is the geometrical meaning of the constant $k$ in the law of sines, $\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c}=k$ in hyperbolic geometry? I know the meaning of the constant only in Euclidean and spherical geometry.
 A: "k" is the "distance scale," traditionally taken to be 1, therefore in a space of curvature $-1,$ as the curvature is $-1/k^2.$ In this and other ways, $k$ appears as a sort of imaginary radius. Note the curvature of the ordinary sphere of radius $r$ is $1/r^2.$
Oh, $k$ does NOT appear in the place you indicate in the Law of Sines. Erase it!
If you want to allow other $k,$ the correct Law is
$$ \frac{\sin A}{\sinh(a/k)} =  \frac{\sin B}{\sinh(b/k)} =  \frac{\sin C}{\sinh(c/k)} $$
The actual meaning of $k$ is a relation between curves called horocycles. But, for something easier, the area of a geodesic triangle is its angular defect multiplied by $k^2.$ 
The easiest introduction I know to these matters is MY_ARTICLE
EDIT: evidently Apotema wanted some other geometric number associated with a triangle that gives the same number as the common value in the Law of Sines. I cannot imagine anything understandable that does that. See the article by Milnor on the first 150 years of hyperbolic geometry: MILNOR. There is no nice expression for the volume of a tetrahedron in $\mathbf H^3.$ I've got to think about whether I even know the volume of a geodesic sphere in $\mathbf H^3.$ Had to look it up, $$ V = 2 \, \pi \, ( \, \sinh r \; \cosh r \; \; - \; \; r    ) = \pi \sinh(2r) - 2 \pi r, $$ and that the Taylor series of this around $r=0$ has first term $\frac{4}{3} \pi r^3,$ as is required in the small.
A: As given by Will Jagy, k must be inside the argument:
$$ \frac{\sin A}{\sinh(a/k)} =  \frac{\sin B}{\sinh(b/k)} =  \frac{\sin C}{\sinh(c/k)} $$
This is the Law of Hyperbolic trigonometry where k is the pseudoradius, constant Gauss curvature $K= -1/k^2$. Please also refer to " Pan-geometry", a set of relations mirrored from spherical to hyerbolic, typified by (sin,cos) -> (sinh,cosh).. in Roberto Bonola's book on Non-euclidean Geometry.
There is nothing imaginary about pseudoradius.It is as real,palpable and solid as the radius of sphere in spherical trigonometry, after hyperbolic geometry has been so firmly established.
I wish practice of using $ K=-1$ should be done away with,always using $ K = -1/k^2 $ or $ K = -1/a^2 $instead. 
