Volumes of Polytopes in Spaces of Constant Curvature

  • Nikolay AbrosimovEmail author
  • Alexander Mednykh
Part of the Fields Institute Communications book series (FIC, volume 70)


We overview volume calculations for polyhedra in Euclidean, spherical and hyperbolic spaces. We prove the Sforza formula for the volume of an arbitrary tetrahedron in H 3 and S 3. We also present some results, which provide a solution for the Seidel problem on the volume of non-Euclidean tetrahedron. Finally, we consider a convex hyperbolic quadrilateral inscribed in a circle, horocycle or one branch of equidistant curve. This is a natural hyperbolic analog of the cyclic quadrilateral in the Euclidean plane. We find several versions of the Brahmagupta formula for the area of such quadrilateral. We also present a formula for the area of a hyperbolic trapezoid.


Volumes of polyhedra Constant curvature spaces Sforza formula Seidel problem Brahmagupta formula 

Subject Classifications

Primary 51M20 Secondary 51M25 51M09 52B15 



The work is supported by Russian Foundation for Basic Research (grants 12-01-00210, 12-01-33058, 12-01-31006 and 13-01-00513) and Council for Grants of President of the Russian Federation (grants MK-4447.2012.1 and SS-921.2012.1).

The authors would like to express their gratitude to a referee for careful consideration of the paper and useful remarks and suggestions.


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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirsk State UniversityNovosibirskRussia

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