Trirectangular tetrahedron

A trirectangular tetrahedron can be constructed by a coordinate octant and a plane crossing all 3 axes away from the origin, like:
x>0
y>0
z>0
and x/a+y/b+z/c<1

In geometry, a trirectangular tetrahedron is a tetrahedron where all three face angles at one vertex are right angles. That vertex is called the right angle of the trirectangular tetrahedron and the face opposite it is called the base. The three edges that meet at the right angle are called the legs and the perpendicular from the right angle to the base is called the altitude of the tetrahedron.

Metric formulas

If the legs have lengths a, b, c, then the trirectangular tetrahedron has the volume

$V={\frac {abc}{6}}.$

The altitude h satisfies^[1]

${\frac {1}{h^{2}}}={\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}+{\frac {1}{c^{2}}}.$

The area $T_{0}$ of the base is given by^[2]

$T_{0}={\frac {abc}{2h}}.$

De Gua's theorem

If the area of the base is $T_{0}$ and the areas of the three other (right-angled) faces are $T_{1}$, $T_{2}$ and $T_{3}$, then

$T_{0}^{2}=T_{1}^{2}+T_{2}^{2}+T_{3}^{2}.$

This is a generalization of the Pythagorean theorem to a tetrahedron.

See also

• Disphenoid
• Goursat tetrahedron
• Orthocentric tetrahedron
• Schläfli orthoscheme
• Standard simplex

References

1. Eves, Howard Whitley, "Great moments in mathematics (before 1650)", Mathematical Association of America, 1983, p. 41.
2. Gutierrez, Antonio, "Right Triangle Formulas",

External links

• Weisstein, Eric W. "Trirectangular tetrahedron". MathWorld.