De Gua's theorem

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tetrahedron with a right-angle corner in O

De Gua's theorem is a three-dimensional analog of the Pythagorean theorem and named for Jean Paul de Gua de Malves.

If a tetrahedron has a right-angle corner (like the corner of a cube), then the square of the area of the face opposite the right-angle corner is the sum of the squares of the areas of the other three faces.

 A_{ABC}^2 = A_{\color {blue} ABO}^2+A_{\color {green} ACO}^2+A_{\color {red} BCO}^2

The Pythagorean theorem and de Gua's theorem are special cases (n = 2, 3) of a general theorem about n-simplices with a right angle corner.

Jean Paul de Gua de Malves (1713–85) published the theorem in 1783, but around the same time a slightly more general version was published by another French mathematician, Tinseau d'Amondans (1746–1818), as well. However the theorem had been known much earlier to Johann Faulhaber (1580–1635) and René Descartes (1596–1650).[1][2]


  1. ^ Weisstein, Eric W., "de Gua's theorem", MathWorld.
  2. ^ Hans-Bert Knoop: Ausgewählte Kapitel zur Geschichte der Mathematik. Lecture Notes (University of Düsseldorf), p. 55 (§ 4 Pythagoreische n-Tupel, p. 50-65) (German)


Further reading[edit]

  • Kheyfits, Alexander (2004). "The Theorem of Cosines for Pyramids". The College Mathematics Journal (Mathematical Association of America) 35 (5): 385–388. JSTOR 4146849.  Proof of de Gua's theorem and of generalizations to arbitrary tetrahedra and to pyramids.