Ingeometry, atrirectangular tetrahedron is atetrahedron where all threefaceangles at onevertex areright angles. That vertex is called theright angle orapex of the trirectangular tetrahedron and the face opposite it is called thebase. The threeedges that meet at the right angle are called thelegs and the perpendicular from the right angle to the base is called thealtitude of the tetrahedron (analogous to thealtitude of a triangle).

An example of a trirectangular tetrahedron is atruncatedsolid figure near the corner of acube or anoctant at the origin ofEuclidean space.Kepler discovered the relationship between the cube, regular tetrahedron and trirectangular tetrahedron.^[1]

Only the bifurcating graph of the${\displaystyle B_3}$affine Coxeter group has a Trirectangular tetrahedron fundamental domain.

If the legs have lengthsa, b, c, then the trirectangular tetrahedron has thevolume^[2]

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

The altitudeh satisfies^[3]

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

The area${\displaystyle T_0}$ of the base is given by^[4]

${\displaystyle T_0=\frac{abc}{2h}.}$

Thesolid angle at the right-angled vertex, from which the opposite face (the base) subtends anoctant, has measureπ/2 steradians, one eighth of the surface area of aunit sphere.

If thearea of the base is${\displaystyle T_0}$ and the areas of the three other (right-angled) faces are${\displaystyle T_1}$,${\displaystyle T_2}$ and${\displaystyle T_3}$, then

${\displaystyle T_0^2=T_1^2+T_2^2+T_3^2.}$

This is a generalization of thePythagorean theorem to a tetrahedron.

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

    a        b        c        d        e        f

   240      117       44      125      244      267   275      252      240      348      365      373   480      234       88      250      488      534   550      504      480      696      730      746   693      480      140      500      707      843   720      351      132      375      732      801   720      132       85      157      725      732   792      231      160      281      808      825   825      756      720     1044     1095     1119   960      468      176      500      976     1068  1100     1008      960     1392     1460     1492  1155     1100     1008     1492     1533     1595  1200      585      220      625     1220     1335  1375     1260     1200     1740     1825     1865  1386      960      280     1000     1414     1686  1440      702      264      750     1464     1602  1440      264      170      314     1450     1464

Notice that some of these are multiples of smaller ones. Note alsoA031173.

Trirectangular tetrahedrons with integer faces${\displaystyle T_c,T_a,T_b,T_0}$ and altitudeh exist, e.g.${\displaystyle a=42,b=28,c=14,T_c=588,T_a=196,T_b=294,T_0=686,h=12}$ without or${\displaystyle a=156,b=80,c=65,T_c=6240,T_a=2600,T_b=5070,T_0=8450,h=48}$ with coprime${\displaystyle a,b,c}$.

• Irregular tetrahedra
• Standard simplex
• Euler Brick

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

• Tetrahedra

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