Isosceles Tetrahedron


An isosceles tetrahedron is a nonregular tetrahedron in which each pair of opposite polyhedron edges are equal, i.e., a^'=a, b^'=b, and c^'=c, so that all triangular faces are congruent. Isosceles tetrahedra are therefore isohedra.

The volume of an isosceles triangle is given by


(Klee and Wagon 1991, p. 205).

A tetrahedron is isosceles iff the sum of the face angles at each polyhedron vertex is 180 degrees, and iff its insphere and circumsphere are concentric (Altshiller-Court 1979, p. 97).

The only way for all the faces of a general tetrahedron to have the same perimeter or to have the same area is for them to be fully congruent, in which case the tetrahedron is isosceles.

The circumradius of an isosceles tetrahedron can be found by plugging in the volume of a general tetrahedron into the relationship


where V is the volume and Delta is the area of a triangle with side lengths a^2, b^2, and c^2 to obtain


See also

Circumsphere, Disphenoid, Insphere, Isohedron, Isosceles Triangle, Regular Tetrahedron, Tetrahedron

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Cite this as:

Weisstein, Eric W. "Isosceles Tetrahedron." From MathWorld--A Wolfram Web Resource.

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