# Is there a neat formula for the volume of a tetrahedron on the surface of $S^3$?

There is a nice formula for the area of a triangle on the surface of the 2-dimensional sphere; If the triangle is the intersection of three half spheres, and has angles $\alpha$, $\beta$ and $\gamma$, and we normalize the area of the whole sphere to be $4\pi$ then the area of the triangle is $$\alpha + \beta + \gamma - \pi.$$ The proof is a cute application of inclusion-exclusion of three sets, and involves the fact that the area we want to calculate appears on both sides of the equation, but with opposite signs.

However, when trying to copy the proof to the three dimensional sphere the parity goes the wrong way and you get 0=0.

Is there a simple formula for the volume of the intersection of four half-spheres of $S^3$ in terms of the 6 angles between the four bounding hyperplanes?

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On the volume of a hyperbolic and spherical tetrahedron, by Murakami and Yano. The volume is obtained as a linear combination of dilogarithms and squares of logarithms. The origin of their formula is really interesting: Asymptotics of quantum $6j$ symbols. (These asymptotics have also been studied by many other people: D. Thurston, Roberts, Woodward, Frohman, Kania-Bartoszynska, etc.)