The volume of a compact hyperbolic antiprism

We consider a compact hyperbolic antiprism. It is a convex polyhedron with 2n vertices in the hyperbolic space H³. This polyhedron has a symmetry group S2n generated by a mirror-rotational symmetry of order 2n, i.e. rotation to the angle π/n followed by a reflection. We establish necessary and sufficient conditions for the existence of such polyhedra in H³. Then we find relations between their dihedral angles and edge lengths in the form of a cosine rule. Finally, we obtain exact integral formulas expressing the volume of a hyperbolic antiprism in terms of the edge lengths.