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
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(1)
|
Specifying the tetrahedron by the three polyhedron edge vectors ,
, and
from a given polyhedron vertex, the volume is
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(2)
|
If the edge between vertices and
is of length
, then the volume
is given by the
Cayley-Menger determinant
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(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
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(4)
|
involving the six dihedral angles (Dostor; Lee). This is a generalization of the law of cosines to the
tetrahedron. Furthermore, for any ,
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(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
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(6)
|
This is a generalisation of Pythagoras's theorem which also applies to higher dimensional simplices (F. M. Jackson).
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
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(7)
|
(P. Kaeser).
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, Granville, Xu and Yau, Guy).
There are a number of interesting and unexpected theorems on the properties of general (i.e., not necessarily regular) tetrahedron (Altshiller-Court). 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). It follows that any plane passing through a bimedian of a tetrahedron bisects the volume of the tetrahedron (Altshiller-Court).
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
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(8)
|
(Crelle; von Staudt; Rouché and Comberousse; Altshiller-Court).
Let be the area of the spherical triangle formed
by the
th face of a tetrahedron in a sphere of
radius
, and let
be the
angle subtended by edge
. Then
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(9)
|
as shown by J.-P. Gua de Malves. The above formula provides the means to calculate the
solid angle subtended by the vertex of a regular tetrahedron by
substituting
(the dihedral angle). Consequently,
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(10)
|
or approximately 0.55129 steradians.