Arkiv för Matematik

  • Ark. Mat.
  • Volume 51, Number 1 (2013), 99-123.

Volume formula for a ℤ2-symmetric spherical tetrahedron through its edge lengths

Alexander Kolpakov, Alexander Mednykh, and Marina Pashkevich

Full-text: Open access


The present paper considers volume formulæ, as well as trigonometric identities, that hold for a tetrahedron in 3-dimensional spherical space of constant sectional curvature +1. The tetrahedron possesses a certain symmetry: namely rotation of angle π in the middle points of a certain pair of its skew edges.


Supported by the Swiss National Science Foundation no. 200020-113199/1, RFBR no. 09-01-00255 and RFBR no. 10-01-00642.

Article information

Ark. Mat., Volume 51, Number 1 (2013), 99-123.

Received: 2 August 2010
Revised: 25 February 2011
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2011 © Institut Mittag-Leffler


Kolpakov, Alexander; Mednykh, Alexander; Pashkevich, Marina. Volume formula for a ℤ 2 -symmetric spherical tetrahedron through its edge lengths. Ark. Mat. 51 (2013), no. 1, 99--123. doi:10.1007/s11512-011-0148-2.

Export citation


  • Abrosimov, N. V., Godoy-Molina, M. and Mednykh, A. D., On the volume of a spherical octahedron with symmetries, Sovrem. Mat. Prilozh. 60Algebra (2008), 3–12 (Russian). English transl.: J. Math. Sci. (N.Y.) 161 (2009), 1–10.
  • Alekseevsky, D. V., Vinberg, E. B. and Solodovnikov, A. S., Geometry of spaces of constant curvature, in Geometry II, Encyclopedia of Mathematical Sciences 29, pp. 1–138, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988 (Russian). English transl.: Springer, Berlin–Heidelberg–New York, 1993.
  • Cho, Y. and Kim, H., On the volume formula for hyperbolic tetrahedra, Discrete Comput. Geom. 22 (1999), 347–366.
  • Derevnin, D. A. and Mednykh, A. D., A formula for the volume of a hyperbolic tetrahedron, Uspekhi Mat. Nauk 60 (2005), 159–160 (Russian). English transl.: Russian Math. Surveys 60 (2005), 346–348.
  • Derevnin, D. A. and Mednykh, A. D., The volume of the Lambert cube in spherical space, Mat. Zametki 86 (2009), 190–201 (Russian). English transl.: Math. Notes 86 (2009), 176–186.
  • Derevnin, D. A., Mednykh, A. D. and Pashkevich, M. G., On the volume of a symmetric tetrahedron in hyperbolic and spherical spaces, Sibirsk. Mat. Zh. 45 (2004), 1022–1031 (Russian). English transl.: Siberian Math. J. 45 (2004), 840–848.
  • Kashaev, R. M., The hyperbolic volume of knots from the quantum dilogarithm, Lett. Math. Phys. 39 (1997), 269–275.
  • Kellerhals, R., On the volume of hyperbolic polyhedra, Math. Ann. 285 (1989), 541–569.
  • Luo, F., On a problem of Fenchel, Geom. Dedicata 64 (1997), 277–282.
  • Luo, F., 3-dimensional Schläfli formula and its generalization, Commun. Contemp. Math. 10 (2008), 835–842.
  • Mednykh, A. D., On hyperbolic and spherical volumes for link cone-manifolds, in Kleinian Groups and Hyperbolic 3-Manifolds, London Math. Soc. Lecture Notes Ser. 229, pp. 145–163, Cambridge Univ. Press, Cambridge, 2003.
  • Milnor, J., The Schläfli differential equality, in Collected Papers. I. Geometry, pp. 281–295, Publish or Perish, Houston, TX, 1994.
  • Murakami, J., On the volume formulas for a spherical tetrahedron, Preprint, 2010.
  • Murakami, J. and Ushijima, A., A volume formula for hyperbolic tetrahedra in terms of edge lengths, J. Geom. 83 (2005), 153–163.
  • Murakami, J. and Yano, M., On the volume of hyperbolic and spherical tetrahedron, Comm. Anal. Geom. 13 (2005), 379–400.
  • Prasolov, V., Problèmes et théorèmes d’algèbre linéaire, Enseignement des mathématiques, Cassini, Paris, 2008.
  • Sabitov, I. Kh., The volume as a metric invariant of polyhedra, Discrete Comput. Geom. 20 (1998), 405–425.
  • Schläfli, L., On the multiple integral $\iint\dots\int{d}x \,{d}y\,\dots\,{d}z$ whose limits are p1=a1x+b1y+…+h1z, p2>0, …, pn>0 and x2+y2+…+z2<1, Quart. J. Math. 2 (1858), 269–300; 3 (1860), 54–68; 97–108.