A trirectangular tetrahedron with its base shown in green and its apex as a solid black disk. It can be constructed by a coordinate octant and a plane crossing all 3 axes away from the origin (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 or apex 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 (analogous to the altitude of a triangle).

Kepler's drawing of a regular tetrahedron inscribed in a cube (on the left), and one of the four trirectangular tetrahedra that surround it (on the right), filling the cube.

An example of a trirectangular tetrahedron is a truncated solid figure near the corner of a cube or an octant at the origin of Euclidean space. Kepler discovered the relationship between the cube, regular tetrahedron and trirectangular tetrahedron.

Only the bifurcating graph of the B 3 {\displaystyle B_{3}} affine Coxeter group has a Trirectangular tetrahedron fundamental domain.

Metric formulas

If the legs have lengths x, y, z then the trirectangular tetrahedron has the volume

V = x y z 6 . {\displaystyle V={\frac {xyz}{6}}.}

The altitude h satisfies

1 h 2 = 1 x 2 + 1 y 2 + 1 z 2 . {\displaystyle {\frac {1}{h^{2}}}={\frac {1}{x^{2}}}+{\frac {1}{y^{2}}}+{\frac {1}{z^{2}}}.}

The area T 0 {\displaystyle T_{0}} of the base is given by

T 0 = x y z 2 h . {\displaystyle T_{0}={\frac {xyz}{2h}}.}

The solid angle at the right-angled vertex, from which the opposite face (the base) subtends an octant, has measure π/2 steradians, one eighth of the surface area of a unit sphere.

De Gua's theorem

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

T 0 2 = T 1 2 + T 2 2 + T 3 2 . {\displaystyle T_{0}^{2}=T_{1}^{2}+T_{2}^{2}+T_{3}^{2}.}

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

Integer solution

Integer edges

Trirectangular tetrahedrons with integer legs a , b , c {\displaystyle a,b,c} and sides d = b 2 + c 2 , e = a 2 + c 2 , f = a 2 + b 2 {\displaystyle d={\sqrt {b^{2}+c^{2}}},e={\sqrt {a^{2}+c^{2}}},f={\sqrt {a^{2}+b^{2}}}} of the base triangle exist, e.g. a = 240 , b = 117 , c = 44 , d = 125 , e = 244 , f = 267 {\displaystyle a=240,b=117,c=44,d=125,e=244,f=267} (discovered 1719 by Halcke). Here are a few more examples with integer legs and sides.

Notice that some of these are multiples of smaller ones. Note also A031173.

Integer faces

Trirectangular tetrahedrons with integer faces T c , T a , T b , T 0 {\displaystyle T_{c},T_{a},T_{b},T_{0}} and altitude h exist, e.g. a = 42 , b = 28 , c = 14 , T c = 588 , T a = 196 , T b = 294 , T 0 = 686 , h = 12 {\displaystyle a=42,b=28,c=14,T_{c}=588,T_{a}=196,T_{b}=294,T_{0}=686,h=12} without or a = 156 , b = 80 , c = 65 , T c = 6240 , T a = 2600 , T b = 5070 , T 0 = 8450 , h = 48 {\displaystyle a=156,b=80,c=65,T_{c}=6240,T_{a}=2600,T_{b}=5070,T_{0}=8450,h=48} with coprime a , b , c {\displaystyle a,b,c}.

See also

External links