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Isosceles Tetrahedron

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IsoscelesTetrahedron

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

V=sqrt(((a^2+b^2-c^2)(a^2+c^2-b^2)(b^2+c^2-a^2))/(72))
(1)

(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

6RV=Delta,
(2)

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

R=sqrt(1/8(a^2+b^2+c^2)).
(3)

See also

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

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References

--. Arts. 7 and 8 in "Géométrie. Mémoire sur le tétraèdre, présentant la solution de diverses questions proposées dans les Annales." Ann. math. 1, 353-367, 1810-1811.Altshiller-Court, N. "The Isosceles Tetrahedron." §4.6b in Modern Pure Solid Geometry. New York: Chelsea, pp. 94-101 and 300, 1979.Biddle, D. Problem 14684. Math. Questions and Solutions from the Educational Times 75, 133-136, 1901.Biddle, D. Mathesis, p. 91, 1931.Brown, B. H. "Undergraduate Mathematics Clubs: Theorem of Bang. Isosceles Tetrahedra." Amer. Math. Monthly 33, 224-226, 1926.Chefik-Bey. Nouv. Ann. 19, 403, 1880.Gentry, E. "Exercices sur le tétraèdre." Nouvelles ann. de math. 37, 223-225, 1878.Honsberger, R. "A Theorem of Bang and the Isosceles Tetrahedron." Ch. 9 in Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 90-97, 1976.Jacobi, C. F. A. In Swinden, J. H. Elemente. p. 457, 1834.Klee, V. and Wagon, S. Old and New Unsolved Problems in Plane Geometry and Number Theory, rev. ed. Washington, DC: Math. Assoc. Amer., 1991.Lemoine, E. "Quelques théorèmes sur les tétraèdres dont les arêtes opposées sont égales deux a deux, et solution de la question 1272." Nouvelle ann. de math. 39, 133-138, 1880.Lemoine, E. Z. Math. u. Physik 29, 321, 1884.Monge, G. Corresp. sur l'École Polytech., pp. 1-6, 1809.Morley, F. "Problem 12032." Math. Questions and Solutions from the Educational Times 61, 26-27, 1894.

Cite this as:

Weisstein, Eric W. "Isosceles Tetrahedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IsoscelesTetrahedron.html

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