My answer to the "Favorite Equations"equations" question was the Pythagorean Theoremtheorem for Rightright-Corner Tetrahedracorner tetrahedra:
where A$A$, B$B$, C$C$ are the areas of the "leg-faces" and D$D$ is the area of the "hypotenuse-face".
Edit to add a couple of references
"The Lawslaws of Cosinescosines for Nonnon-Euclidean Tetrahedra"tetrahedra" (PDFpdf file) (by me) derives a Law of Cosines for which each non-Euclidean Pythagorean Theorem is a special case. I do not know if these results exist elsewhere in the literature. (BTW: There's a lot of unnecessary equation manipulation shown; I was using this as an opportunity to learn LaTeX. :)
A passage in above jumps into a discussion of "pseudofaces" by making passing reference to the Euclidean case. You can read about Euclidean pseudofaces and how I find them useful here:
"Heron-like Resultsresults for [Euclidean] Tetrahedral Volume"tetrahedral volume" (by me).
Here, $L^\star$ is half of Catalan's Constantconstant, and $H^\star$ is also known in the literature. (It is, for instance, the maximum of the Clausen function $\mathrm{Cl}_2$.)
If there's going to be a Pythagorean Theoremtheorem for Hyperbolic Simpliceshyperbolic simplices, then it must apply to this case, ideally relating these values in the non-Euclidean Pythagorean tradition: