Is there a volume formula for hyperbolic tetrahedron

Question

Is there a volume formula for hyperbolic truncated tetrahedron? Which seems looks like Yu. Cho and H. Kim, Discrete Comput. Geom. 22, 347–366 (1999).

Thanks.

Answer 1

Look for the volume of a simplex. Suppose you have $n+1$ affine independent column vectors $v_1,v_2,\cdots,v_{n+1}$ in $\mathbb R^n$ then the simplex of these points(convex hull of them) has volume:

$$\det \begin{bmatrix} 1 & 1 & \cdots & 1 \\ v_1 & v_2 & \cdots & v_{n+1} \\ \end{bmatrix} $$

Now you can split hyperbolic truncated tetrahedron to some simplex.