Tetrahedron

In general, a tetrahedron is a polyhedron with four sides.

If all faces are congruent, the tetrahedron is known as an isosceles tetrahedron. If all faces are congruent to an equilateral triangle, then the tetrahedron is known as a regular tetrahedron (although the term "tetrahedron" without further qualification is often used to mean "regular tetrahedron"). A tetrahedron having a trihedron all of the face angles of which are right angles is known as a trirectangular tetrahedron.

A general (not necessarily regular) tetrahedron, defined as a convex polyhedron consisting of four (not necessarily identical) triangular faces can be specified by its polyhedron vertices as , where , ..., 4. Then the volume of the tetrahedron is given by

(1)

Specifying the tetrahedron by the three polyhedron edge vectors , , and from a given polyhedron vertex, the volume is

(2)

If the edge between vertices and is of length , then the volume is given by the Cayley-Menger determinant

(3)

Consider an arbitrary tetrahedron with triangles , , , and . Let the areas of these triangles be , , , and , respectively, and denote the dihedral angle with respect to and for by . Then the four face areas are connected by

(4)

involving the six dihedral angles (Dostor 1905, pp. 252-293; Lee 1997). This is a generalization of the law of cosines to the tetrahedron. Furthermore, for any ,

(5)

where is the length of the common edge of and (Lee 1997).

Given a right-angled tetrahedron with one apex where all the edges meet orthogonally and where the face opposite this apex is denoted , then

(6)

This is a generalisation of Pythagoras's theorem which also applies to higher dimensional simplices (F. M. Jackson, pers. comm., Feb. 20, 2006).

Let be the set of edges of a tetrahedron and the power set of . Write for the complement in of an element . Let be the set of triples such that span a face of the tetrahedron, and let be the set of , so that and . In , there are therefore three elements which are the pairs of opposite edges. Now define , which associates to an edge of length the quantity , , which associates to an element the product of for all , and , which associates to the sum of for all . Then the volume of a tetrahedron is given by

(7)

(P. Kaeser, pers. comm.).

The analog of Gauss's circle problem can be asked for tetrahedra: how many lattice points lie within a tetrahedron centered at the origin with a given inradius (Lehmer 1940, Granville 1991, Xu and Yau 1992, Guy 1994).

There are a number of interesting and unexpected theorems on the properties of general (i.e., not necessarily regular) tetrahedron (Altshiller-Court 1979). If a plane divides two opposite edges of a tetrahedron in a given ratio, then it divides the volume of the tetrahedron in the same ratio (Altshiller-Court 1979, p. 89). It follows that any plane passing through a bimedian of a tetrahedron bisects the volume of the tetrahedron (Altshiller-Court 1979, p. 90).

Let the vertices of a tetrahedron be denoted , , , and , and denote the side lengths , , , , , and . Then if denotes the area of the triangle with sides of lengths , , and , the volume and circumradius of the tetrahedron are related by the beautiful formula

(8)

(Crelle 1821, p. 117; von Staudt 1860; Rouché and Comberousse 1922, pp. 568-576 and 643-664; Altshiller-Court 1979, p. 249).

Let be the area of the spherical triangle formed by the th face of a tetrahedron circumscribed in a sphere of radius and let be the angle subtended by edge . Then

(9)

as shown by J.-P. Gua de Malves around 1740 or 1783 (Hopf 1940). The above formula provides the means to calculate the solid angle subtended from a vertex by the opposite face of a regular tetrahedron by substituting (the dihedral angle) into the above formula. Consequently,

			(10)
			(11)

or approximately 0.55129 steradians.

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