TOPICS
Search

Trirectangular Tetrahedron

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook
TrirectangularTetrahedron
TrirectangularTetraWire

A trirectangular tetrahedron is 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)

Conversely, let x=sqrt(b^2+c^2), y=sqrt(a^2+c^2), and z=sqrt(a^2+b^2) be the side lengths of the base. Solving these relations for the three perpendicular edges gives

a=sqrt(1/2(y^2+z^2-x^2))
(2)
b=sqrt(1/2(x^2+z^2-y^2))
(3)
c=sqrt(1/2(x^2+y^2-z^2)).
(4)

A triangle is the base of a nondegenerate trirectangular tetrahedron iff it is an acute triangle. The volume of the trirectangular tetrahedron is

V=1/6abc.
(5)

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)).
(6)

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
(7)

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),
(8)

with s defined by (1).

See also

Acute Triangle, de Gua's Theorem, Tetrahedron, Trihedron

Explore with Wolfram|Alpha

References

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

Referenced on Wolfram|Alpha

Trirectangular Tetrahedron

Cite this as:

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

Subject classifications