Chapter 42

Tetrahedra

42.1 The isosceles tetrahedron

An isosceles tetrehedron is one whose four faces are congruent trian-
gles. Given a triangleABC, construct its anticomplimentary triangle
A′B′C′by drawing lines through the vertices parallel to their opposite
sides. Fold along the sides of the given triangle to bring the verticesA′,
B′,C′into a pointD, forming an isosceles tetrahedronABCD. Every
isosceles tetrahedron arises from any one of its faces in this way. We
may therefore ask for the volume of the isosceles tetrahedronABCDin
terms of the side lengths of triangleABC.

A

B C

B′=D

A′=D

C′=D A

B C

D

L

To compute the volume of a tetrahedron, we would drop the perpen-
dicular from a vertex to its opposite face (of areaΔ) to determine the
heighthon this face. The volume of the tetrahedron is thenV =^13 Δh.
For an isosceles tetrahedron, the position of this perpendicular foot is
clearly the same for the four faces.