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Trirectangular Tetrahedron

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TrirectangularTetrahedron
TrirectangularTetraWire

A tetrahedron having a trihedron all of the face angles of which are right angles. The face opposite the vertex of the right angles is called the base. If the edge lengths bounding the trihedral angle are a, b, and c, then the side lengths of the base are given by sqrt(a^2+b^2), sqrt(a^2+c^2), and sqrt(b^2+c^2), and so has semiperimeter

s=1/2(sqrt(a^2+b^2)+sqrt(a^2+c^2)+sqrt(b^2+c^2)).
(1)

The volume of the trirectangular tetrahedron is

V=1/6abc.
(2)

Using Heron's formula, the surface area is therefore

S=1/2(ab+ac+bc+sqrt(a^2b^2+a^2c^2+b^2c^2)).
(3)

Let Delta_(XYZ) be the area of the triangle with vertices X, Y, and Z. The remarkable de Gua's theorem

Delta_(ABC)^2=Delta_(OAB)^2+Delta_(OAC)^2+Delta_(OBC)^2
(4)

then follows from the identity

s(s-sqrt(a^2+b^2))(s-sqrt(a^2+c^2))(s-sqrt(b^2+c^2))=1/4(a^2b^2+a^2c^2+b^2c^2),
(5)

with s defined by (1).

See also

de Gua's Theorem, Tetrahedron, Trihedron

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References

Altshiller-Court, N. "The Trirectangular Tetrahedron." §4.6a in Modern Pure Solid Geometry. New York: Chelsea, pp. 91-94, 1979.

Cite this as:

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

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