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
- De Gua's theorem
- Integer solution
- Perfect body
- Integer edges
- Integer faces
- See also
- References
- External links

Only the bifurcating graph of the Affine Coxeter group has a Trirectangular tetrahedron fundamental domain.

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

The altitude *h* satisfies^{ [1] }

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

If the area of the base is and the areas of the three other (right-angled) faces are , and , then

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

The area of the base (a,b,c) is always (Gua) an irrational number. Thus a trirectangular tetrahedron with integer edges is never a perfect body. The trirectangular bipyramid (6 faces, 9 edges, 5 vertices) built from these trirectangular tetrahedrons and the related left-handed ones connected on their bases have rational edges, faces and volume, but the inner space-diagonal between the two trirectangular vertices is still irrational. The later one is the double of the *altitude* of the trirectangular tetrahedron and a rational part of the (proved)^{ [3] } irrational space-diagonal of the related *Euler-brick* (bc, ca, ab).

Trirectangular tetrahedrons with integer legs and sides of the base triangle exist, e.g. (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 A031173.

Trirectangular tetrahedrons with integer faces and altitude *h* exist, e.g. without or with coprime .

In mathematics, a **square root** of a number *x* is a number *y* such that *y*^{2} = *x*; in other words, a number *y* whose *square* (the result of multiplying the number by itself, or *y* ⋅ *y*) is *x*. For example, 4 and −4 are square roots of 16, because 4^{2} = (−4)^{2} = 16. Every nonnegative real number *x* has a unique nonnegative square root, called the *principal square root*, which is denoted by where the symbol is called the *radical sign* or *radix*. For example, the principal square root of 9 is 3, which is denoted by because 3^{2} = 3 ⋅ 3 = 9 and 3 is nonnegative. The term (or number) whose square root is being considered is known as the *radicand*. The radicand is the number or expression underneath the radical sign, in this case 9.

In geometry, a **tetrahedron**, also known as a **triangular pyramid**, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.

A **triangle** is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices *A*, *B*, and *C* is denoted .

A **right triangle** or **right-angled triangle** (British), or more formally an **orthogonal triangle**, formerly called a **rectangled triangle**, is a triangle in which one angle is a right angle or two sides are perpendicular. The relation between the sides and other angles of the right triangle is the basis for trigonometry.

In geometry, an **equilateral triangle** is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a **regular triangle**.

In geometry, **Heron's formula**, named after Hero of Alexandria, gives the area of a triangle when the lengths of all three sides are known. Unlike other triangle area formulae, there is no need to calculate angles or other distances in the triangle first.

In English outside North America, a convex quadrilateral in Euclidean geometry, with at least one pair of parallel sides, is referred to as a **trapezium** ; in American and Canadian English this is usually referred to as a **trapezoid**. The parallel sides are called the *bases* of the trapezoid. The other two sides are called the *legs* if they are not parallel; otherwise, the trapezoid is a parallelogram, and there are two pairs of bases). A *scalene trapezoid* is a trapezoid with no sides of equal measure, in contrast with the special cases below.

In mathematics, an ** nth root** of a number

In mathematics, a **quadratic irrational number** is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numbers. Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their least common denominator, a quadratic irrational is an irrational root of some quadratic equation with integer coefficients. The quadratic irrational numbers, a subset of the complex numbers, are algebraic numbers of degree 2, and can therefore be expressed as

The **square root of 2** is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as or , and is an algebraic number. Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property.

In geometry, a **median** of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side. Every triangle has exactly three medians, one from each vertex, and they all intersect each other at the triangle's centroid. In the case of isosceles and equilateral triangles, a median bisects any angle at a vertex whose two adjacent sides are equal in length.

In geometry, a **rhombohedron** is a three-dimensional figure with six faces which are rhombi. It is a special case of a parallelepiped where all edges are the same length. It can be used to define the rhombohedral lattice system, a honeycomb with rhombohedral cells. A cube is a special case of a rhombohedron with all sides square.

**High-leg delta** is a type of electrical service connection for three-phase electric power installations. It is used when both single and three-phase power is desired to be supplied from a three phase transformer. The three-phase power is connected in the delta configuration, and the center point of one phase is grounded. This creates both a split-phase single phase supply and three-phase. It is called "orange leg" because the wire is color-coded orange. By convention, the high leg is usually set in the center lug in the involved panel, regardless of the L1-L2-L3 designation at the transformer.

In geometry, a **disphenoid** is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same shape are **sphenoid**, **bisphenoid**, **isosceles tetrahedron**, **equifacial tetrahedron**, **almost regular tetrahedron**, and **tetramonohedron**.

In geometry, the **spiral of Theodorus** is a spiral composed of right triangles, placed edge-to-edge. It was named after Theodorus of Cyrene.

In mathematics, the values of the trigonometric functions can be expressed approximately, as in , or exactly, as in . While trigonometric tables contain many approximate values, the exact values for certain angles can be expressed by a combination of arithmetic operations and square roots.

In geometry, a **pentagon** is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.

In mathematics, the **Pythagorean theorem**, or **Pythagoras' theorem**, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the legs *a*, *b* and the hypotenuse *c*, often called the **Pythagorean equation**:

An **integer triangle** or **integral triangle** is a triangle all of whose sides have lengths that are integers. A **rational triangle** can be defined as one having all sides with rational length; any such rational triangle can be integrally rescaled to obtain an integer triangle, so there is no substantive difference between integer triangles and rational triangles in this sense. However, other definitions of the term "rational triangle" also exist: In 1914 Carmichael used the term in the sense that we today use the term Heronian triangle; Somos uses it to refer to triangles whose ratios of sides are rational; Conway and Guy define a rational triangle as one with rational sides and rational angles measured in degrees—in which case the only rational triangle is the rational-sided equilateral triangle.

An **acute triangle** is a triangle with three acute angles. An **obtuse triangle** is a triangle with one obtuse angle and two acute angles. Since a triangle's angles must sum to 180° in Euclidean geometry, no Euclidean triangle can have more than one obtuse angle.

- ↑ Eves, Howard Whitley, "Great moments in mathematics (before 1650)",
*Mathematical Association of America*, 1983, p. 41. - ↑ Gutierrez, Antonio, "Right Triangle Formulas",
- ↑ Walter Wyss, "No Perfect Cuboid", arXiv : 1506.02215

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