Some 3D Generalizations of Viviani's Theorem

Theorem 1
The sum of the distances from a point P to the faces of a tetrahedron with faces of equal area, is constant.

As pointed out in De Villiers (1999, p. 150; 2013) and Kawasaki et al (2005), Viviani's theorem generalizes to any tetrahedron with faces of equal area. More-over, as shown in Brown (1926) and Andreescu & Gelca (2009) such a tetrahedron has congruent (acute-angled) faces and is called a disphenoid. A dynamic net that folds up to produce a disphenoid is shown above and consists of constructing the midpoints of the sides of any acute-angled triangle to form four congruent triangles.
Challenge Can you explain (prove) why the result is true? Only if stuck, go to Disphenoid Viviani Proof.

Theorem 1 generalizes to any polyhedron with faces of equal area, and therefore not only includes the other four regular polyhedra, but also irregular ones like the hexagonal bipyramid.

Theorem 1 Converse
If the sum of the distances from a point P to the faces of a tetrahedron is constant, then the tetrahedron has faces of equal area (i.e. is a disphenoid). Can you explain (prove) why the result is true? Only if stuck, go to Disphenoid Viviani Converse Proof.

Theorem 2
If a prism is created by the orthogonal translation in space of an equilateral polygon, equi-angled polygon or a 2n-gon with opposite sides parallel, then the sum of the distances from a point P to the faces of the prism, is constant.

Proof
In the case of prisms formed by the orthogonal translation of these polygons, the distances from P to the side faces formed by the translation all lie in the same plane parallel to the original polygon, and the intersection of this plane with the prism produces a polygon congruent to the original. More-over, the translated polygon and its image are parallel; hence the result.

For a 2n-gon with opposite sides parallel, the translation need not be orthogonal, but can be oblique. A manipulable 3D sketch illustrates the result for a hexagon ABCDEF with opposite sides parallel, which has been translated by the blue vector to form a prism. Since opposite faces are all parallel, the sum of the distances between each pair of faces is constant - as shown by the distance measurements for the respective pairs of green and yellow faces (Results in green & orange). In the Cabri 3D software, one can rotate the solid, or drag point P, any vertices of the hexagon or the vector to explore it dynamically.

Since the sum of distances from a point to a pair of parallel faces is constant, Viviani's theorem also generalizes to any polyhedron with opposite faces parallel, and therefore not only includes regular ones like the octahedron, dodecahedron and the icosahedron, but also irregular ones like the cuboctahedron.

Lastly, note that the results hold even when P is outside the polyhedron or outside a pair of parallel planes, provided we regard distances respectively falling completely outside the polyhedron or outside a pair planes as negative; in other words using directed distances (or equivalently, vectors). However, Cabri 3D does not measure 'negative' distances, so dragging P outside will appear to no longer give a constant sum.

It has kindly been brought to my attention in December 2013 by Li Zhou from Polk State College, Winter Haven, USA, that the Theorem 2 above immediately follows from Theorem 1 proved in a paper of his - see (Zhou, 2012).

Download Cabri 3D files
Disphenoid Viviani
Parallel-hexagon Face Prism Viviani

References
Andreescu, T. & Gelca, R. (2009). Mathematical Olympiad Challenges. Birkhauser, second edition, 2009, pp. 30-31.
Brown, B. H. (1926). Theorem of Bang. Isosceles tetrahedra, American Mathematical Monthly, April, pp. 224-226.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Emeryville: Key Curriculum Press.
De Villiers, M. (2013). 3D Generalisations of Viviani's theorem. The Mathematical Gazette, Vol 97, No 540 (November), pp. 441-445.
Kawasaki, K-I.; Yagi, Y. & Yanagawa, K. (2005). On Viviani's theorem in three dimensions. The Mathematical Gazette, Vol. 89, No. 515 (Jul), pp. 283-287.
Zhou, L. (2012). Viviani Polytopes and Fermat Points. College Mathematics Journal, 43 (4), pp. 309–312.

Related Links
Distances in an Equilateral Triangle (Viviani's theorem) (Rethinking Proof activity)
2D Generalizations of Viviani's Theorem
Further generalizations of Viviani's Theorem (Using equi-inclined lines)
Clough's Theorem (a variation of Viviani) and some Generalizations
A theorem by Wares
Carnot's (or Bottema's) Perpendicularity Theorem & Some Generalizations
Water Supply: Four Towns (Rethinking Proof activity: intro perpendicular bisectors)
Triangle Altitudes (Rethinking Proof activity)
Light Ray in a Triangle (Rethinking Proof activity)
Maximising the Area of the 3rd Pedal Triangle in Neuberg's theorem
Power Lines of a Triangle
Desargues' Theorem
The surprising 3D parallelo-hexagon

Some External Links
Viviani's theorem (Wikipedia)
Viviani's Theorem (Cut The Knot)

Free Downloads of 3D Software
Cabri Express (for Windows)
Cabri Express (for Mac)
GeoGebra 3D Calculator (3D grapher
Desmos 3D Calculator (Online 3D grapher)

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Created by Michael de Villiers, 22 August 2012; updated 6 Dec 2025.