There is a trigonometry of tetrahedra (separate from spherical trigonometry others mention), although it doesn't deal with solid angles. It's dimensionally-enhanced in a way that relates areas and dihedral angles. I call this field "hedronometry", and I've referenced some key results on Math.SE from time to time. For instance, I discuss the Law(s!) of Cosines here; for completeness, I'll include them here:
Given a tetrahedron $OABC$, with faces of area $W$, $X$, $Y$, $Z$ opposite respective vertices $O$, $A$, $B$, $C$, and with $\angle PQ$ denoting the dihedral angle along edge $\overline{PQ}$, we have
The First Law of Cosines
$$W^2 = X^2 + Y^2 + Z^2 - 2 Y Z \cos \angle OA - 2 Z X \cos \angle OB - 2 X Y \cos \angle OC$$
In particular, for a "right-corner" tetrahedron with hypotenuse-face $W$, we have a Pythagorean Theorem, aka de Gua's Theorem:
$$W^2 = X^2 + Y^2 + Z^2$$
The Second Law of Cosines
$$\begin{eqnarray*}
W^2 + X^2 - 2 W X \cos \angle BC &= H^2 =& Y^2 + Z^2 - 2 Y Z \cos \angle OA \\
W^2 + Y^2 - 2 W Y \cos \angle CA &= J^2 =& Z^2 + X^2 - 2 Z X \cos \angle OB \\
W^2 + Z^2 - 2 W Z \cos \angle AB &= K^2 =& X^2 + Y^2 - 2 X Y \cos \angle OC
\end{eqnarray*}$$
Here, $H$, $J$, $K$ are (what I call) "pseudo-faces" of the tetrahedron. They're related to projections of the tetrahedron in planes parallel to opposite edges. In any case, combining these Laws yields this tidy result:
The Sum-of-Squares Identity
$$W^2 + X^2 + Y^2 + Z^2 = H^2 + J^2 + K^2$$
Note that, whereas a triangle is determined by its three side-lengths, a tetrahedron —which admits six degrees of freedom— is not determined by its four face-areas. The three pseudo-faces fill the gap (and the Sum-of-Squares Identity prevents them from over-filling it!), so "hedronometry" is about how all the metric properties of a tetrahedron can be expressed in terms of face and pseudo-face areas.
You can read more about this stuff on my Hedronometry pages (which desperately need a style overhaul). The more-recent entries deal with the hedronometry of hyperbolic space, so you'll probably want to go back in time a bit for Euclidean stuff. I intend, someday, to compile that information into a unified document.
