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Isosceles Tetrahedron

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IsoscelesTetrahedron

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

The volume of an isosceles tetrahedron 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 nonregular tetrahedron is isosceles iff the sum of the angles of its faces 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 congruent triangles, in which case the solid is a regular tetrahedron or an isosceles tetrahedron.

The circumradius of an isosceles tetrahedron can be found by substituting its volume into the relationship

6RV=Delta,
(2)

where V is the volume and Delta is the triangle area of an auxiliary 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. Mathesis, p. 91, 1931.Biddle, D. Problem 14684. Math. Questions and Solutions from the Educational Times 75, 133-136, 1901.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.

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Isosceles Tetrahedron

Cite this as:

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

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