I have updated my note, “Hedronometric Formulas for a Hyperbolic Tetrahedron” (PDF), with a brand new formula for the volume of an arbitrary tetrahedron in terms of its face and pseudo-face areas. (See Section 8.3.)

The formula isn’t the monolithic and symmetric counterpart to Derevnin-Mednykh I’ve been seeking, but it’s a start. It’s complicated enough that I won’t attempt to render it here.

The **Open Question**: As one might expect, the formula involves an integral. One of the limits of integration is the subject of a Conjecture. Again, the notion is too complicated to describe here, but the gist is that I *believe* that, by appropriately assigning names to the faces (and pseudo-faces), we guarantee that the lower limit is simply one-quarter of a particular pseudo-face area. (If I’m mistaken, then that limit is a less-obvious root of a trigonometric equation.) Numerical experiments in Mathematica suggest that the conjecture is true, but I don’t have even non-constructive proof. (Nevermind that the conjecture wouldn’t really be helpful without a practical way to determine what an “appropriate assignment” of names would be.) When (if?) a properly symmetric formula is finally discovered, the order of face names won’t matter at all; but for now, it makes for an irksome little wrinkle in the formula.