True or False Quickie: The volume of a tetrahedron is determined by the areas of its faces. _______________________________________________
False. There exists a face area preserving continuous deformation that alters the volume. Start with a regular tetrahedron. Perform a dilatation along the mutual perpendicular of two opposite edges, while simultaneously performing a dilatation in the transverse plane so as to preserve the face areas. This is possible because all four faces are inclined with respect to that mutual perpendicular by the same angle. The volume is not preserved, in particular the volume goes to zero as the tetrahedron is squeezed flat.
an easy way to see this is to take four alternating vertices of a rectangular parallelepiped; then all four faces are congruent. i've heard this figure called an "isosceles tetrahedron". if one dimension of the rectangular parallelepiped is very small, then the volume of the tetrahedron is very small, but the area of the faces is not necessarily small. specifically, if the rectangular parallelepiped has dimensions a x b x c , then the volume of the tetrahedron is abc / 3 , and the area of the congruent faces is sqrt(a^2 b^2 + b^2 c^2 + c^2 a^2) / 2 . mike