geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a trirectangular tetrahedron is a

tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

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 Altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum (geodesy), datum and a point or object. The exact definition and reference datum varies according to the context (e.g., aviation, geometr ...

of a triangle). An example of a trirectangular tetrahedron is a truncated solid figure near the corner of a

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

or an octant at the origin of

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

Kepler Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws of p ...

discovered the relationship between the cube, regular tetrahedron and trirectangular tetrahedron. Only the bifurcating graph of the

B_3

affine Coxeter group has a Trirectangular tetrahedron fundamental domain.

Metric formulas

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

volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...

V=\frac.

The altitude ''h'' satisfies :

\frac=\frac+\frac+\frac.

The area

T_0

of the base is given byGutierrez, Antonio, "Right Triangle Formulas"
/ref> :

T_0=\frac.

The solid angle at the right-angled vertex, from which the opposite face (the base) subtends an octant, has measure /2

steradian The steradian (symbol: sr) or square radian is the unit of solid angle in the International System of Units (SI). It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. A solid angle in the fo ...

s, one eighth of the surface area of a

unit sphere In mathematics, a unit sphere is a sphere of unit radius: the locus (mathematics), set of points at Euclidean distance 1 from some center (geometry), center point in three-dimensional space. More generally, the ''unit -sphere'' is an n-sphere, -s ...

De Gua's theorem

If the

area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...

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 In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

to a tetrahedron.

Integer solution

Integer edges

Trirectangular tetrahedrons with integer legs

a,b,c

and sides

d=\sqrt, e=\sqrt, f=\sqrt

of the base triangle exist, e.g.

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 also .

Integer faces

Trirectangular tetrahedrons with integer faces

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

without or

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

References

External links

*{{MathWorld , title = Trirectangular tetrahedron , urlname = TrirectangularTetrahedron Tetrahedra