This dynamic 3D applet does not work on Internet Explorer, and works best on a relatively new version of the free browser Firefox - click link to download latest version. It also requires the downloading & installation of the free Cabri 3D Plug In, available at Windows (4 Mb) or Mac OS (13.4 Mb).

# 3D Generalizations of Viviani's Theorem

Basic manipulation: 1) Right click (or Ctrl + click) and drag to rotate the whole figure (glassball).
2) Click to select and hold down the left button to drag P - the constant sum of the distances to the faces is shown under Result. The shape of the tetrahedron can be changed by dragging any of the vertices A, B or C.
Or click Summary of manipulation to open & resize a separate window with instructions.

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 earlier pointed out in De Villiers (1999 & 2003, p. 150), 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. 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.

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). Rethinking Proof with Geometer's Sketchpad. Emeryville: Key Curriculum Press.

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. The manipulable 3D sketch above 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). 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.

Update: Read my paper 3D Generalizations of Viviani's theorem in the November 2013 issue of The Mathematical Gazette.

It has also kindly been brought to my attention in December 2013 by Li Zhou from Polk State College, Winter Haven, USA that in a 2012 paper of his Viviani Polytopes and Fermat Points in the September 2012 issue of the College Mathematics Journal that my Theorem 2 above immediately follows from his Theorem 1.