Disphenoid Viviani Proof

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

Proof: Since a point P inside a disphenoid divides it into four smaller tetrahedra so that if A represents the area of each face and hi the four heights, then ⅓A(h1 + h2 + h3 + h4) = V <=> h1 + h2 + h3 + h4 = 3V/A, where V represents the volume of the disphenoid. Therefore, since V and A are constant, the sum of the distances from P to the faces is constant. QED.
(Note: An interesting property of the disphenoid is also apparent from the above, namely, that the four heights (altitudes) of the disphenoid are equal since the volume is constant and each face has the same area, and the volume calculated from each face, has to be the same.)

{Formula for volume of pyramid
1) A simple explanation/derivation of the volume formula for a general pyramid (of which a tetrahedron is a special case) is given at: Simple Explanations for Area and Volume Formulas.
2) Another simple explanation, and informal proof, for the formula of a pyramid is given at: Volume of a Pyramid and a Cone.
3) This project from a grade 6 class with video and pictures shows How to split a cube into 3 identical pyramids with square bases.
4) Make your own paper models of the three pyramids in no. 3 at Template for paper models and fit together to form a cube.}