In this paper the authors give a closed formula for volumes of generic hyperbolic tetrahedra in terms of edge lengths.

**Volume formulae for hyperbolic tetrahedra**

For complex numbers z, a_{1}, a_{2}, . . . , a_{6}, define a complex valued function U = U(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, z) as follows:

where Li_{2} is the dilogarithm function

Then define z+ and z- by,

where

Define a complex-valued function V(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6},) as follows:

For the tetrahedron

in this case:

where

and

is the Gram matrix defined in terms of dihedral angles.

Then

and

where,

and

G_{l} is the Gram matrix in terms of edge lengths defined as

** Related articles**

- Numerical work with sagemath 24: 6j Symbols and non-eucledean Tetrahedra
- Closure constraints for hyperbolic tetrahedra
- Classical 6j-symbols and the tetrahedron

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