We know that, given an n-dimensional Euclidean simplex, for all $1\leq i,j,k,l\leq n+1$, we have(law of sines)$$\frac{A\_i A\_j}{A\_k A\_l}=\frac{c\_{ij}}{c\_{kl}}$$(from *Elementary Formulas for a Hyperbolic Tetrahedron*)
And in <http://www.emis.de/journals/JIPAM/images/106_03_JIPAM/106_03.pdf> it mentioned some inequalities for simplexes in Euclidean space.


I wonder if there is a formula about law of sines and cosines for hyperbolic n-simplexes which only have terms: side lengths, area, dihedral angles.


Thanks.



Is there a law of cosine for n-dimensional hyperbolic simplex
Yes, something like that is proved in the paper by Simon Kokkendorff:


*Kokkendorff, Simon L.*, [**Polar duality and the generalized law of sines**](http://dx.doi.org/10.1007/s00022-006-1858-7), J. Geom. 86, No. 1-2, 140-149 (2006). [ZBL1115.51010](https://zbmath.org/?q=an:1115.51010).


It would be too cumbersome to define everything here, but the paper is very nicely written.


