De Gua's theorem

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De Gua's theorem

Summary

De Gua's theorem is a remarkable extension of the Pythagorean theorem from two to three dimensions. Instead of relating the lengths of the sides of a right triangle, it relates the areas of the faces of a right tetrahedron—a pyramid whose three edges meeting at one vertex are mutually perpendicular. The theorem states that the square of the area of the face opposite the right-angled vertex equals the sum of the squares of the areas of the other three faces, mirroring the familiar relationship ($a^2+b^2=c^2$). 

First published by the French mathematician Jean Paul de Gua de Malves in 1783, the result was later found to have been known earlier by René Descartes and Johann Faulhaber. Beyond its elegant geometric interpretation, the theorem is connected to Heron’s formula and has inspired broader generalizations to higher-dimensional simplices, where similar relationships hold between the volumes of orthogonal projections. Today, De Gua’s theorem is regarded as one of the most beautiful examples of how classical geometric ideas naturally extend into higher dimensions. 

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