Siberian Mathematical Journal

, Volume 47, Issue 4, pp 687–695 | Cite as

Elementary formulas for a hyperbolic tetrahedron

  • A. D. Mednykh
  • M. G. Pashkevich


We derive some elementary formulas expressing the relation between the dihedral angles and edge lengths of a tetrahedron in hyperbolic space.


hyperbolic tetrahedron n-dimensional hyperbolic simplex law of sines law of cosines 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cho Yu. and Kim H., “On the volume formula for hyperbolic tetrahedra,” Discrete Comput. Geom., 22, No. 3, 347–366 (1999).zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Murakami J. and Yano M., On the Volume of Hyperbolic Tetrahedron [Preprint] (2001). Available at http: // Scholar
  3. 3.
    Ushijima A., A Volume Formula for Generalized Hyperbolic Tetrahedra [Preprint] (2002). Available at http: // Scholar
  4. 4.
    Derevnin D. A. and Mednykh A. D., “A formula for the volume of a hyperbolic tetrahedron,” Russian Math. Surveys, 60, No. 2, 346–348 (2005).zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Coolidge J. L., The Elements of Non-Euclidean Geometry, Clarendon Press, Oxford (1909).Google Scholar
  6. 6.
    Derevnin D. A., Mednykh A. D., and Pashkevich M. G., “On the volume of a symmetric tetrahedron in hyperbolic and spherical spaces,” Siberian Math. J., 45, No. 5, 840–848 (2004).zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Rivin I., “A multidimensional theorem of sines.” Available at
  8. 8.
    Ratcliffe J. G., Foundations of Hyperbolic Manifolds, Springer-Verlag, New York (1994).Google Scholar
  9. 9.
    Fenchel W., Elementary Geometry in Hyperbolic Space, de Gruyter, Berlin; New York (1989).Google Scholar
  10. 10.
    Eriksson F., “The law of sines for tetrahedra and n-simplices,” Geom. Dedicata, 7, No. 1, 71–80 (1978).zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Prasolov V. V., Problems and Theorems in Linear Algebra, Amer. Math. Soc., Providence, RI (1994).Google Scholar
  12. 12.
    Prasolov V. V. and Sossinsky A. B., Knots, Links, Braids, and 3-Manifolds [in Russian], MTsNMO, Moscow (1997).Google Scholar
  13. 13.
    Buser P., Geometry and Spectra of Compact Riemann Surfaces, Birkhauser, Boston; Basel; Berlin (1992) (Progress in Mathematics; 106).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • A. D. Mednykh
    • 1
  • M. G. Pashkevich
    • 2
  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Siberian State Transport UniversityNovosibirskRussia

Personalised recommendations