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{{distinguish|tetrahedroid}}
{{for|the academic journal|Tetrahedron (journal)}}
{{Reg polyhedra db|Reg polyhedron stat table|T}}
In [[geometry]], a '''tetrahedron''' (plural: '''tetrahedra''' or '''tetrahedrons''') is a [[polyhedron]] composed of four [[triangular]] [[Face (geometry)|faces]], three of which meet at each corner or [[vertex (geometry)|vertex]]. It has six [[Edge (geometry)|edges]] and four vertices. The tetrahedron is the simplest of all the ordinary [[convex polytope|convex polyhedra]] and the only  one that has four faces.<ref name="MW">{{MathWorld |urlname=Tetrahedron |title=Tetrahedron }}</ref>

The tetrahedron is the three-dimensional case of the more general concept of a [[Euclidean geometry|Euclidean]] [[simplex]].

The tetrahedron is one kind of [[pyramid (geometry)|pyramid]], which is a polyhedron with a flat [[polygon]] base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid".

Like all [[convex polyhedra]], a tetrahedron can be folded from a single sheet of paper. It has two such [[net (polyhedron)|nets]].<ref name="MW" />

For any tetrahedron the vertices lie on a sphere (called the [[circumsphere]]) while another sphere (the [[insphere]]) just touches the tetrahedron's faces.

==Regular tetrahedron==
A '''regular tetrahedron''' is one in which all four faces are [[equilateral triangle]]s. It has been known since antiquity and is one of the five regular [[Platonic solid]]s.

In a regular tetrahedron, not only are all its faces the same size and shape (congruent) but so are all its vertices and edges.

Together with the [[regular octahedron]], these two solids can be packed alternately to fill space, however regular tetrahedra alone cannot fill space.
[[File:Tetrahedrons cannot fill space..PNG|thumb|300px|Five tetrahedra are laid flat on a plane, with the highest 3-dimensional points marked as 1, 2, 3, 4, and 5. These points are then attached to each other and a thin volume of empty space is left, where the five edge angles do not quite meet.]]

The regular tetrahedron is self-dual, which means that its [[Dual polyhedron|dual]] is another regular tetrahedron. The [[Polyhedral compound|compound]] figure comprising two such dual tetrahedra form a [[stellated octahedron]] or stella octangula.

===Formulas for a regular tetrahedron===<!--[[Tetrahedral angle]] redirects here-->
The following Cartesian coordinates define the four vertices of a tetrahedron with edge length 2, centered at the origin:
:(±1, 0, −1/√2)
:(0, ±1, 1/√2)

Another set of coordinates are based on an [[Alternation (geometry)|alternated]] [[cube]] with edge length 2. The tetrahedron in this case has edge length <math>2\sqrt{2}</math>. Inverting these coordinates generates the dual tetrahedron, and the pair together represent the stellated octahedron. (Note that if the pair are merged unordered, they give the vertices of the original cube.)
:Tetrahedon: (1,1,1), (1,−1,−1), (−1,1,−1), (−1,−1,1)
:Dual tetrahedron: (−1,−1,−1), (−1,1,1), (1,−1,1), (1,1,−1)

[[File:Вписанный тетраэдр.svg|thumb|right|300px|Regular tetrahedron ABCD and its circumscribed sphere]]
For a regular tetrahedron of edge length ''a'':
{|class="wikitable"
|width=300|Face area
|align=center|<math>A_0={\sqrt{3}\over4}a^2\,</math>
|-
|[[area (mathematics)|Surface area]]<ref name="Cox">[[Harold Scott MacDonald Coxeter|Coxeter, Harold Scott MacDonald]]; ''[[Regular Polytopes (book)|Regular Polytopes]]'', Methuen and Co., 1948, Table I(i)</ref>
|align=center|<math>A=4\,A_0={\sqrt{3}}a^2\,</math>
|-
|Height of pyramid<ref>Köller, Jürgen, [http://www.mathematische-basteleien.de/tetrahedron.htm "Tetrahedron"], Mathematische Basteleien, 2001</ref>
|align=center|<math>H={\sqrt{6}\over3}a=\sqrt{2\over3}\,a\,</math>
|-
|[[Volume]]<ref name="Cox" />
|align=center|<math>V={1\over3} A_0h ={\sqrt{2}\over12}a^3={a^3\over{6\sqrt{2}}}\,</math>
|-
|Face-vertex-edge angle
|align=center|<math>\arccos\left({1 \over \sqrt{3}}\right) = \arctan(\sqrt{2})\,</math><br>(approx. 54.7356°)
|-
|[[Table of polyhedron dihedral angles|Face-edge-face angle]]<ref name="Cox" />
|align=center|<math>\arccos\left({1 \over 3}\right) = \arctan(2\sqrt{2})\,</math><br>(approx. 70.5288°)
|-
|[[Tetrahedral molecular geometry|Edge central angle]],<ref>[http://maze5.net/?page_id=367 "Angle Between 2 Legs of a Tetrahedron"], Maze5.net</ref> <ref>[http://pubs.acs.org/doi/abs/10.1021/ed022p145?journalCode=jceda8 Valence Angle of the Tetrahedral Carbon Atom] W.E. Brittin, J. Chem. Educ., 1945, 22 (3), p 145</ref> known as the ''tetrahedral angle''
|align=center|<math>\arccos\left({-1\over3}\right ) = 2\arctan(\sqrt{2})\,</math><br>(approx. 109.4712°)
|-
|[[Solid angle]] at a vertex subtended by a face
|align=center|<math>\arccos\left({23\over27}\right)</math><br>(approx. 0.55129 [[steradian]]s)
|-
|Radius of [[circumsphere]]<ref name="Cox" />
|align=center|<math>R={\sqrt{6}\over4}a=\sqrt{3\over8}\,a\,</math>
|-
|Radius of [[insphere]] that is tangent to faces<ref name="Cox" />
|align=center|<math>r={1\over3}R={a\over\sqrt{24}}\,</math>
|-
|Radius of [[midsphere]] that is tangent to edges<ref name="Cox" />
|align=center|<math>r_M=\sqrt{rR}={a\over\sqrt{8}}\,</math>
|-
|Radius of [[Exsphere (polyhedra)|exspheres]]
|align=center|<math>r_E={a\over\sqrt{6}}\,</math>
|-
|Distance to exsphere center from a vertex
|align=center|<math>d_{VE}={\sqrt{6}\over2}a=\sqrt{3\over2}\,a\,</math>
|}

Note that with respect to the base plane the [[slope]] of a face (<math>\scriptstyle 2 \sqrt{2}</math>) is twice that of an edge (<math>\scriptstyle \sqrt{2}</math>), corresponding to the fact that the ''horizontal'' distance covered from the base to the [[Apex (geometry)|apex]] along an edge is twice that along the [[Median (geometry)|median]] of a face. In other words, if ''C'' is the [[centroid]] of the base, the distance from ''C'' to a vertex of the base is twice that from ''C'' to the midpoint of an edge of the base. This follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other (see [[centroid#Proof that the centroid of a triangle divides each median in the ratio 2:1|proof]]).

===Isometries of the regular tetrahedron===
[[Image:Symmetries of the tetrahedron.svg|thumb|400px|The proper rotations, (order-3 rotation on a vertex and face, and order-2 on 2 edges) and reflection plane (through 2 faces and one edge) in the symmetry group of the regular tetrahedron]]
The vertices of a [[cube]] can be grouped into two groups of four, each forming a regular tetrahedron (see above, and also [[:Image:Tetraeder animation with cube.gif|animation]], showing one of the two tetrahedra in the cube). The [[Symmetry in mathematics|symmetries]] of a regular tetrahedron correspond to half of those of a cube: those that map the tetrahedra to themselves, and not to each other.

The tetrahedron is the only Platonic solid that is not mapped to itself by [[point inversion]].

The regular tetrahedron has 24 isometries, forming the [[symmetry group]] '''T<sub>d</sub>''', [3,3], (*332), isomorphic to the [[symmetric group]], ''S''<sub>4</sub>. They can be categorized as follows:
* '''T''', [3,3]<sup>+</sup>, (332) is isomorphic to [[alternating group]], ''A''<sub>4</sub> (the identity and 11 proper rotations) with the following [[conjugacy class]]es (in parentheses are given the permutations of the vertices, or correspondingly, the faces, and the [[Quaternions and spatial rotation|unit quaternion representation]]):
** identity (identity; 1)
** rotation about an axis through a vertex, perpendicular to the opposite plane, by an angle of ±120°: 4 axes, 2 per axis, together {{nowrap|8 ((1 2 3)}}, etc.; {{nowrap|(1 ± ''i'' ± ''j'' ± ''k'') / 2)}}
** rotation by an angle of 180° such that an edge maps to the opposite edge: {{nowrap|3 ((1 2)(3 4)}}, etc.; {{nowrap|''i'', ''j'', ''k''}})
* reflections in a plane perpendicular to an edge: 6
* reflections in a plane combined with 90° rotation about an axis perpendicular to the plane: 3 axes, 2 per axis, together 6; equivalently, they are 90° rotations combined with inversion ('''x''' is mapped to −'''x'''): the rotations correspond to those of the cube about face-to-face axes

===Orthogonal projections of the regular tetrahedron===
The regular ''tetrahedron'' has two special [[orthogonal projection]]s, one centered on a vertex or equivalently on a face, and one centered on an edge. The first corresponds to the A<sub>2</sub> [[Coxeter plane]].

{| class=wikitable
|+ [[Orthogonal projection]]
!Centered by
!Face/vertex
!Edge
|- align=center
!Image
|[[File:3-simplex t0 A2.svg|100px]]
|[[File:3-simplex t0.svg|100px]]
|- align=center
!Projective<br>symmetry
![3]
![4]
|}

==Other special cases==
[[File:Tetrahedral subgroup tree tidy.svg|thumb|Tetrahedral symmetry subgroup relations]]
{{Further|#Isometries of irregular tetrahedra}}
An '''isosceles tetrahedron''', also called a [[disphenoid]], is a tetrahedron where all four faces are [[Congruence (geometry)|congruent]] triangles.  A '''space-filling tetrahedron''' packs with congruent copies of itself to tile space, like the [[disphenoid tetrahedral honeycomb]].

In a [[trirectangular tetrahedron]] the three face angles at one vertex are [[right angle]]s. If all three pairs of opposite edges of a tetrahedron are [[perpendicular]], then it is called an [[orthocentric tetrahedron]]. When only one pair of opposite edges are perpendicular, it is called a '''semi-orthocentric tetrahedron'''. An '''isodynamic tetrahedron''' is one in which the [[cevian]]s that join the vertices to the [[Incircle and excircles of a triangle|incenters]] of the opposite faces are [[Concurrent lines|concurrent]], and an '''isogonic tetrahedron''' has concurrent cevians that join the vertices to the points of contact of the opposite faces with the [[inscribed sphere]] of the tetrahedron.

===Isometries of irregular tetrahedra===
The isometries of an irregular (unmarked) tetrahedron depend on the geometry of the tetrahedron, with 7 cases possible. In each case a [[Point groups in three dimensions|3-dimensional point group]] is formed. Two other isometries (C<sub>3</sub>, [3]<sup>+</sup>), and (S<sub>4</sub>, [2<sup>+</sup>,4<sup>+</sup>]) can exist if the face or edge marking are included. Tetrahedral diagrams are included for each type below, with edges colored by isometric equivalence, and are gray colored for unique edges.
{| class=wikitable width=800
!colspan=4|Tetrahedron name
!rowspan=3|Edge<BR>Equivalence<BR>diagram
!rowspan=3|Description
|- align=center
!colspan=4|[[List of spherical symmetry groups|Symmetry]]
|- align=center
![[Schönflies notation|Schön.]]
![[Coxeter notation|Cox.]]
![[Orbifold notation|Orb.]]
![[Symmetry order|Ord.]]
|- align=center
!colspan=4|Regular Tetrahedron
|rowspan=2|[[File:Regular tetrahedron diagram.png|60px]]
|rowspan=2 align=left|<center>Four '''equilateral''' triangles</center><P>It forms the symmetry group ''T''<sub>d</sub>, isomorphic to the [[symmetric group]], ''S''<sub>4</sub>.
|- align=center
||''T''<sub>d</sub><BR>''T''||[3,3]<BR>[3,3]<sup>+</sup>||*332<BR>332|| 24<BR>12
|- align=center
!colspan=4|[[Triangular pyramid]]
|rowspan=2|[[File:Isosceles trigonal pyramid diagram.png|60px]]
|rowspan=2 align=left|<center>An '''equilateral''' triangle base and three equal '''isosceles''' triangle sides</center><BR>It gives 6 isometries, corresponding to the 6 isometries of the base. As permutations of the vertices, these 6 isometries are the identity 1, (123), (132), (12), (13) and (23), forming the symmetry group ''C''<sub>3v</sub>, isomorphic to the [[symmetric group]], ''S''<sub>3</sub>.
|- align=center
||''C''<sub>3v</sub><BR>C<sub>3</sub> ||[3]<BR>[3]<sup>+</sup> || *33<BR>33 ||6<BR>3

|- align=center
!colspan=4|Mirrored sphenoid
|rowspan=2|[[File:Sphenoid diagram.png|60px]]
|rowspan=2 align=left|<center>Two equal '''scalene''' triangles with a common base edge</center><BR>This has two pairs of equal edges (1,3), (1,4) and (2,3), (2,4) and otherwise no edges equal. The only two isometries are 1 and the reflection (34), giving the group ''C''<sub>s</sub>, also isomorphic to the [[cyclic group]], '''Z'''<sub>2</sub>.
|-align=center
|''C''<sub>s</sub><BR>=''C''<sub>1h</sub><BR>=''C''<sub>1v</sub>||[&nbsp;] ||*|| 2

|-align=center
!colspan=4|Irregular tetrahedron<BR>(No symmetry)
|rowspan=2|[[File:Scalene tetrahedron diagram.png|60px]]
|rowspan=2 align=left|<center>Four unequal triangles</center><P>Its only isometry is the identity, and the symmetry group is the [[trivial group]].
|-align=center
|C<sub>1</sub>||[&nbsp;]<sup>+</sup>||1||1

|-align=center
|colspan=6 bgcolor="f0a0a0" |'''[[Disphenoid]]s''' (Four equal triangles)
|-align=center
!colspan=4|[[Tetragonal disphenoid]]
|rowspan=2|[[File:Tetragonal disphenoid diagram.png|60px]]
|rowspan=2 align=left|<center>Four equal '''isosceles''' triangles</center><P>It has 8 isometries. If edges (1,2) and (3,4) are of different length to the other 4 then the 8 isometries are the identity 1, reflections (12) and (34), and 180° rotations (12)(34), (13)(24), (14)(23) and improper 90° rotations (1234) and (1432) forming the symmetry group ''D''<sub>2d</sub>.
|-align=center
|''D''<sub>2d</sub><BR>S<sub>4</sub> ||[2<sup>+</sup>,4]<BR>[2<sup>+</sup>,4<sup>+</sup>] ||2*2<BR>2×|| 8<BR>4
|-align=center
!colspan=4|[[Rhombic disphenoid]]
|rowspan=2|[[File:Rhombic disphenoid diagram.png|60px]]
|rowspan=2 align=left|<center>Four equal '''scalene''' triangles</center><P>It has 4 isometries. The isometries are 1 and the 180° rotations (12)(34), (13)(24), (14)(23). This is the [[Klein four-group]] ''V''<sub>4</sub> or '''Z'''<sub>2</sub><sup>2</sup>, present as the point group ''D<sub>2</sub>''.
|-align=center
|''D<sub>2</sub>'' ||[2,2]<sup>+</sup> ||222||4

|-align=center
|colspan=6 bgcolor="f0a0a0" |'''Generalized disphenoids''' (2 pairs of equal triangles)
|-align=center
!colspan=4|[[Digonal disphenoid]]
|rowspan=2|[[File:Digonal disphenoid diagram.png|60px]]
|rowspan=2 align=left|<center>Two pairs of equal '''isosceles''' triangles</center><P>This gives two opposite edges (1,2) and (3,4) that are perpendicular but different lengths, and then the 4 isometries are 1, reflections (12) and (34) and the 180° rotation (12)(34). The symmetry group is ''C''<sub>2v</sub>, isomorphic to the [[Klein four-group]] ''V''<sub>4</sub>.
|-align=center
|''C''<sub>2v</sub><BR>=''D''<sub>1h</sub> ||[2] ||*22||4

|-align=center
!colspan=4|Phyllic disphenoid
|rowspan=2|[[File:Half-turn tetrahedron diagram.png|60px]]
|rowspan=2 align=left|<center>Two pairs of equal '''scalene''' triangles</center><P>This has two pairs of equal edges (1,3), (2,4) and (1,4), (2,3) but otherwise no edges equal. The only two isometries are 1 and the rotation (12)(34), giving the group ''C''<sub>2</sub> isomorphic to the [[cyclic group]], '''Z'''<sub>2</sub>.
|-align=center
|''C''<sub>2</sub><BR>=''D''<sub>1</sub> ||[2]<sup>+</sup> ||22 ||2
|}

==General properties==

===Volume===
The volume of a tetrahedron is given by the pyramid volume formula:
:<math>V = \frac{1}{3} A_0\,h \,</math>

where ''A''<sub>0</sub> is the area of the base and ''h'' the height from the base to the apex. This applies for each of the four choices of the base, so the distances from the apexes to the opposite faces are inversely proportional to the areas of these faces.

For a tetrahedron with vertices
{{nowrap|'''a''' {{=}} (''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>)}},
{{nowrap|'''b''' {{=}} (''b''<sub>1</sub>, ''b''<sub>2</sub>, ''b''<sub>3</sub>)}},
{{nowrap|'''c''' {{=}} (''c''<sub>1</sub>, ''c''<sub>2</sub>, ''c''<sub>3</sub>)}}, and
{{nowrap|'''d''' {{=}} (''d''<sub>1</sub>, ''d''<sub>2</sub>, ''d''<sub>3</sub>)}}, the volume is {{nowrap|(1/6)·{{!}}[[determinant{{!}}det]]('''a''' − '''d''', '''b''' − '''d''', '''c''' − '''d'''){{!}}}}, or any other combination of pairs of vertices that form a simply connected [[graph theory|graph]]. This can be rewritten using a [[dot product]] and a [[cross product]], yielding
:<math>V = \frac { |(\mathbf{a}-\mathbf{d}) \cdot ((\mathbf{b}-\mathbf{d}) \times (\mathbf{c}-\mathbf{d}))| } {6}.</math>

If the origin of the coordinate system is chosen to coincide with vertex '''d''', then '''d''' = 0, so
:<math>V = \frac { |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})| } {6},</math>

where '''a''', '''b''', and '''c''' represent three edges that meet at one vertex, and {{nowrap|'''a''' · ('''b''' × '''c''')}} is a [[scalar triple product]]. Comparing this formula with that used to compute the volume of a [[parallelepiped]], we conclude that the volume of a tetrahedron is equal to 1/6 of the volume of any parallelepiped that shares three converging edges with it.

The triple scalar can be represented by the following determinants:
:<math>6 \cdot V =\begin{vmatrix}
\mathbf{a} & \mathbf{b} & \mathbf{c}
\end{vmatrix}</math>{{pad|2em}}or{{pad|1em}}<math>6 \cdot V =\begin{vmatrix}
\mathbf{a} \\ \mathbf{b} \\ \mathbf{c}
\end{vmatrix}</math>{{pad|2em}}where{{pad|1em}}<math>\mathbf{a} = (a_1,a_2,a_3) \,</math>{{pad|1em}}is expressed as a row or column vector etc.

Hence
:<math>36 \cdot V^2 =\begin{vmatrix}
\mathbf{a^2} & \mathbf{a} \cdot \mathbf{b} & \mathbf{a} \cdot \mathbf{c} \\
\mathbf{a} \cdot \mathbf{b} & \mathbf{b^2} & \mathbf{b} \cdot \mathbf{c} \\
\mathbf{a} \cdot \mathbf{c} & \mathbf{b} \cdot \mathbf{c} & \mathbf{c^2}
\end{vmatrix}</math>{{pad|1em}}where{{pad|1em}}<math>\mathbf{a} \cdot \mathbf{b} = ab\cos{\gamma}</math>{{pad|1em}}etc.

which gives
:<math>V = \frac {abc} {6} \sqrt{1 + 2\cos{\alpha}\cos{\beta}\cos{\gamma}-\cos^2{\alpha}-\cos^2{\beta}-\cos^2{\gamma}}, \,</math>
where ''α'', ''β'', ''γ'' are the plane angles occurring in vertex '''d'''. The angle ''α'', is the angle between the two edges connecting the vertex '''d''' to the vertices '''b''' and '''c'''. The angle ''β'', does so for the vertices '''a''' and '''c''', while ''γ'', is defined by the position of the vertices '''a''' and '''b'''.

Given the distances between the vertices of a tetrahedron the volume can be computed using the [[Cayley–Menger determinant]]:
:<math>288 \cdot V^2 =
\begin{vmatrix}
  0 & 1        & 1        & 1        & 1        \\
  1 & 0        & d_{12}^2 & d_{13}^2 & d_{14}^2 \\
  1 & d_{12}^2 & 0        & d_{23}^2 & d_{24}^2 \\
  1 & d_{13}^2 & d_{23}^2 & 0        & d_{34}^2 \\
  1 & d_{14}^2 & d_{24}^2 & d_{34}^2 & 0
\end{vmatrix}</math>
where the subscripts <math>i,\,j\in\{1,\,2,\,3,\,4\}</math> represent the vertices {'''a''', '''b''', '''c''', '''d'''} and <math>\scriptstyle d_{ij}</math> is the pairwise distance between them – i.e., the length of the edge connecting the two vertices. A negative value of the determinant means that a tetrahedron cannot be constructed with the given distances. This formula, sometimes called [[Tartaglia's formula]], is essentially due to the painter [[Piero della Francesca]] in the 15th century, as a three dimensional analogue of the 1st century [[Heron's formula]] for the area of a triangle.<ref>[http://www.mathpages.com/home/kmath664/kmath664.htm "Simplex Volumes and the Cayley-Menger Determinant"], MathPages.com</ref>

==== Heron-type formula for the volume of a tetrahedron ====
If ''U'', ''V'', ''W'', ''u'', ''v'', ''w'' are lengths of edges of the tetrahedron (first three form a triangle; ''u'' opposite to ''U'' and so on), then<ref>Kahan, William M.; [http://www.cs.berkeley.edu/~wkahan/VtetLang.pdf "What has the Volume of a Tetrahedron to do with Computer Programming Languages?"], pp. 16–17</ref>
:<math>
\text{volume} = \frac{\sqrt {\,( - a + b + c + d)\,(a - b + c + d)\,(a + b - c + d)\,(a + b + c - d)}}{192\,u\,v\,w}</math>

where

: <math>
\begin{align}
    a & = \sqrt {xYZ} \\ b & = \sqrt {yZX} \\ c & = \sqrt {zXY} \\ d & = \sqrt {xyz} \\ X & = (w - U + v)\,(U + v + w) \\ x & = (U - v + w)\,(v - w + U) \\ Y & = (u - V + w)\,(V + w + u) \\ y & = (V - w + u)\,(w - u + V) \\ Z & = (v - W + u)\,(W + u + v) \\ z & = (W - u + v)\,(u - v + W).
\end{align}
</math>

====Non-Euclidean volume====
For tetrahedra in [[hyperbolic space]] or in three-dimensional [[spherical geometry]], the [[dihedral angle]]s of the tetrahedron determine its shape and hence its volume. In these cases, the volume is given by the [[Murakami–Yano formula]].<ref>{{Citation | last1=Murakami | first1=Jun | last2=Yano | first2=Masakazu | title=On the volume of a hyperbolic and spherical tetrahedron | url= http://intlpress.com/CAG/CAG-v13.php#v13n2| id={{MR|2154824}} | year=2005 | journal=Communications in Analysis and Geometry | issn=1019-8385 | volume=13 | issue=2 | pages=379–400}}
</ref> However, in Euclidean space, scaling a tetrahedron changes its volume but not its dihedral angles, so no such formula can exist.

===Distance between the edges===
Any two opposite edges of a tetrahedron lie on two [[skew lines]], and the distance between the edges is defined as the distance between the two skew lines. Let ''d'' be the distance between the skew lines formed by opposite edges '''a''' and {{nowrap|'''b''' − '''c'''}} as calculated [[Skew lines#Distance between two skew lines|here]]. Then another volume formula is given by
:<math>V = \frac {d |(\mathbf{a} \times \mathbf{(b-c)})| } {6}.</math>

===Properties of a general tetrahedron===
The tetrahedron has many properties analogous to those of a triangle, including an insphere, circumsphere, medial tetrahedron, and exspheres. It has respective centers such as incenter, circumcenter, excenters, [[Spieker circle|Spieker center]] and points such as a centroid. However, there is generally no orthocenter in the sense of intersecting altitudes. The circumsphere of the medial tetrahedron is analogous to the triangle's [[nine-point circle]], but does not generally pass through the base points of the altitudes of the reference tetrahedron.<ref>{{Cite journal |last=Havlicek |first=Hans |last2=Weiß |first2=Gunter |title=Altitudes of a tetrahedron and traceless quadratic forms |journal=[[American Mathematical Monthly]] |volume=110 |issue=8 |pages=679–693 |year=2003 |url=http://www.geometrie.tuwien.ac.at/havlicek/pub/hoehen.pdf |doi=10.2307/3647851 |jstor=3647851 }}</ref>

[[Gaspard Monge]] found a center that exists in every tetrahedron, now known as the ''Monge point'': the point where the six midplanes of a tetrahedron intersect. A midplane is defined as a plane that is orthogonal to an edge joining any two vertices that also contains the centroid of an opposite edge formed by joining the other two vertices. If the tetrahedron's altitudes do intersect, then the Monge point and the orthocenter coincide to give the class of [[orthocentric tetrahedron]].

An orthogonal line dropped from the Monge point to any face meets that face at the midpoint of the line segment between that face's orthocenter and the foot of the altitude dropped from the opposite vertex.

A line segment joining a vertex of a tetrahedron with the [[centroid]] of the opposite face is called a ''median'' and a line segment joining the midpoints of two opposite edges is called a ''bimedian'' of the tetrahedron. Hence there are four medians and three bimedians in a tetrahedron. These seven line segments are all [[Concurrent lines|concurrent]] at a point called the ''centroid'' of the tetrahedron.<ref>Leung, Kam-tim; and Suen, Suk-nam; "Vectors, matrices and geometry", Hong Kong University Press, 1994, pp. 53–54</ref> The centroid of a tetrahedron is the midpoint between its Monge point and circumcenter. These points define the ''Euler line'' of the tetrahedron that is analogous to the [[Euler line]] of a triangle.

The [[nine-point circle]] of the general triangle has an analogue in the circumsphere of a tetrahedron's medial tetrahedron. It is the ''twelve-point sphere'' and besides the centroids of the four faces of the reference tetrahedron, it passes through four substitute ''Euler points'', 1/3 of the way from the Monge point toward each of the four vertices. Finally it passes through the four base points of orthogonal lines dropped from each Euler point to the face not containing the vertex that generated the Euler point.<ref>{{Cite book |last=Outudee |first=Somluck |last2=New |first2=Stephen |title=The Various Kinds of Centres of Simplices |publisher=Dept of Mathematics, Chulalongkorn University, Bangkok |url=http://web.archive.org/web/20090227143222/http://www.math.sc.chula.ac.th/ICAA2002/pages/Somluck_Outudee.pdf }}</ref>

The center ''T'' of the twelve-point sphere also lies on the Euler line. Unlike its triangular counterpart, this center lies 1/3 of the way from the Monge point ''M'' towards the circumcenter. Also, an orthogonal line through ''T'' to a chosen face is coplanar with two other orthogonal lines to the same face. The first is an orthogonal line passing through the corresponding Euler point to the chosen face. The second is an orthogonal line passing through the centroid of the chosen face. This orthogonal line through the twelve-point center lies midway between the Euler point orthogonal line and the centroidal orthogonal line. Furthermore, for any face, the twelve-point center lies at the midpoint of the corresponding Euler point and the orthocenter for that face.

The radius of the twelve-point sphere is 1/3 of the circumradius of the reference tetrahedron.

There is a relation among the angles made by the faces of a general tetrahedron given by <ref>{{cite web |first=Daniel |last=Audet |title=Déterminants sphérique et hyperbolique de Cayley-Menger |url=http://archimede.mat.ulaval.ca/amq/bulletins/mai11/Chronique_note_math.mai11.pdf |publisher=Bulletin AMQ |date=May 2011 }}</ref>
:<math>\begin{vmatrix}  -1 & \cos{(\alpha_{12})} & \cos{(\alpha_{13})} & \cos{(\alpha_{14})}\\
\cos{(\alpha_{12})} & -1 & \cos{(\alpha_{23})} & \cos{(\alpha_{24})} \\
\cos{(\alpha_{13})} & \cos{(\alpha_{23})} & -1 & \cos{(\alpha_{34})} \\
\cos{(\alpha_{14})} & \cos{(\alpha_{24})} & \cos{(\alpha_{34})} & -1 \\ \end{vmatrix} = 0\,</math>

where <math> \alpha_{ij}</math> is the angle between the faces ''i'' and ''j''.

===Geometric relations===
A tetrahedron is a 3-[[simplex]]. Unlike the case of the other Platonic solids, all the vertices of a regular tetrahedron are equidistant from each other (they are the only possible arrangement of four equidistant points in 3-dimensional space).

A tetrahedron is a triangular [[Pyramid (geometry)|pyramid]], and the regular tetrahedron is [[self-dual polyhedron|self-dual]].

A regular tetrahedron can be embedded inside a [[cube (geometry)|cube]] in two ways such that each vertex is a vertex of the cube, and each edge is a diagonal of one of the cube's faces. For one such embedding, the [[Cartesian coordinates]] of the [[Vertex (geometry)|vertices]] are
:(+1, +1, +1);
:(−1, −1, +1);
:(−1, +1, −1);
:(+1, −1, −1).

This yields a tetrahedron with edge-length <math>\scriptstyle 2 \sqrt{2}</math>, centered at the origin. For the other tetrahedron (which is [[dual polyhedron|dual]] to the first), reverse all the signs. These two tetrahedra's vertices combined are the vertices of a cube, demonstrating that the regular tetrahedron is the 3-[[demicube]].

[[File:Compound of two tetrahedra.png|thumb|200px|The [[stella octangula]].]]
The volume of this tetrahedron is 1/3 the volume of the cube. Combining both tetrahedra gives a regular [[polyhedral compound]] called the [[compound of two tetrahedra]] or [[stella octangula]].

The interior of the stella octangula is an [[octahedron]], and correspondingly, a regular octahedron is the result of cutting off, from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e. [[Rectification (geometry)|rectifying]] the tetrahedron).

The above embedding divides the cube into five tetrahedra, one of which is regular. In fact, 5 is the minimum number of tetrahedra required to compose a cube.

Inscribing tetrahedra inside the regular [[Polyhedral compound|compound of five cubes]] gives two more regular compounds, containing five and ten tetrahedra.

Regular tetrahedra cannot [[Honeycomb (geometry)|tessellate space]] by themselves, although this result seems likely enough that [[Aristotle]] claimed it was possible. However, two regular tetrahedra can be combined with an octahedron, giving a [[rhombohedron]] that can tile space.

However, several irregular tetrahedra are known, of which copies can tile space, for instance the [[disphenoid tetrahedral honeycomb]]. The complete list remains an open problem.<ref>{{Cite journal
 | doi = 10.2307/2689983
 | last = Senechal | first = Marjorie | authorlink = Marjorie Senechal
 | title = Which tetrahedra fill space?
 | year = 1981
 | journal = [[Mathematics Magazine]]
 | volume = 54
 | issue = 5
 | pages = 227–243
 | publisher = Mathematical Association of America
 | jstor = 2689983
 | postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}</ref>

If one relaxes the requirement that the tetrahedra be all the same shape, one can tile space using only tetrahedra in many different ways. For example, one can divide an octahedron into four identical tetrahedra and combine them again with two regular ones. (As a side-note: these two kinds of tetrahedron have the same volume.)

The tetrahedron is unique among the [[uniform polyhedron|uniform polyhedra]] in possessing no parallel faces.

===A law of sines for tetrahedra and the space of all shapes of tetrahedra===
[[Image:tetra.png|248px]]

A corollary of the usual [[law of sines]] is that in a tetrahedron with vertices ''O'', ''A'', ''B'', ''C'', we have
:<math>\sin\angle OAB\cdot\sin\angle OBC\cdot\sin\angle OCA = \sin\angle OAC\cdot\sin\angle OCB\cdot\sin\angle OBA.\,</math>

One may view the two sides of this identity as corresponding to clockwise and counterclockwise orientations of the surface.

Putting any of the four vertices in the role of ''O'' yields four such identities, but at most three of them are independent: If the "clockwise" sides of three of them are multiplied and the product is inferred to be equal to the product of the "counterclockwise" sides of the same three identities, and then common factors are cancelled from both sides, the result is the fourth identity.

Three angles are the angles of some triangle if and only if their sum is 180° (π radians). What condition on 12 angles is necessary and sufficient for them to be the 12 angles of some tetrahedron? Clearly the sum of the angles of any side of the tetrahedron must be 180°. Since there are four such triangles, there are four such constraints on sums of angles, and the number of [[Degrees of freedom (statistics)|degrees of freedom]] is thereby reduced from 12 to 8. The four relations given by this sine law further reduce the number of degrees of freedom, from 8 down to not 4 but 5, since the fourth constraint is not independent of the first three. Thus the space of all shapes of tetrahedra is 5-dimensional.<ref>{{Cite journal |title=Is There a "Most Chiral Tetrahedron"? |first1=André |last1=Rassat |first2=Patrick W. |last2=Fowler |journal=Chemistry: A European Journal |volume=10 |issue=24 |pages=6575–6580 |year=2004 |doi=10.1002/chem.200400869 |postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}} }}</ref>

==Related polyhedra and compounds==
A regular tetrahedron can be seen as a triangular [[Pyramid (geometry)|pyramid]].
{{Pyramids}}

A regular tetrahedron can be seen as a degenerate polyhedron, a uniform ''digonal [[antiprism]]'', where base polygons are reduced [[digon]]s.
{{UniformAntiprisms}}

A regular tetrahedron can be seen as a degenerate polyhedron, a uniform dual ''digonal [[trapezohedron]]'', containing 6 vertices, in two sets of colinear edges.
{{Trapezohedra}}

A truncation process applied to the tetrahedron produces a series of [[uniform polyhedra]]. Truncating edges down to points produces the [[octahedron]] as a rectified tetrahedron. The process completes as a birectification, reducing the original faces down to points, and producing the self-dual tetrahedron once again.
{{Tetrahedron family}}

This polyhedron is topologically related as a part of sequence of regular polyhedra with [[Schläfli symbol]]s {3,n}, continuing into the [[Hyperbolic space|hyperbolic plane]].
{{Triangular regular tiling}}

The tetrahedron is topologically related to a series of regular polyhedra and tilings with order-3 [[vertex figure]]s.
{{Order-3 tiling table}}

'''Compounds:'''
<gallery>
Image:CubeAndStel.svg|[[Stella octangula|Two tetrahedra in a cube]]
Image:Compound of five tetrahedra.png|[[Compound of five tetrahedra]]
Image:Compound of ten tetrahedra.png|[[Compound of ten tetrahedra]]
</gallery>

<br style="clear:both"/>

An interesting polyhedron can be constructed from [[Compound of five tetrahedra|five intersecting tetrahedra]]. This [[Polyhedral compound|compound]] of five tetrahedra has been known for hundreds of years. It comes up regularly in the world of [[origami]]. Joining the twenty vertices would form a regular [[dodecahedron]]. There are both [[left-handed]] and [[right-handed]] forms, which are [[mirror image]]s of each other.

==Applications==
[[Image:Ammonium-3D-balls.png|100px|thumb|The [[ammonium]]+ ion is tetrahedral]]
[[Image:4-sided dice 250.jpg|100px|thumb|[[4-sided die]]]]

===Numerical analysis===
In [[numerical analysis]], complicated three-dimensional shapes are commonly broken down into, or [[approximate]]d by, a [[polygon mesh|polygonal mesh]] of irregular [[tetrahedra]] in the process of setting up the equations for [[finite element analysis]] especially in the [[numerical solution]] of [[partial differential equations]]. These methods have wide applications in practical applications in [[computational fluid dynamics]], [[aerodynamics]], [[electromagnetic field]]s, [[civil engineering]], [[chemical engineering]], [[naval architecture|naval architecture and engineering]], and related fields.

===Chemistry===
{{main|Tetrahedral molecular geometry}}
The tetrahedron shape is seen in nature in [[covalent bond]]s of molecules. All sp<sup>3</sup>-hybridized atoms are surrounded by atoms lying in each corner of a tetrahedron. For instance in a [[methane]] molecule (CH<sub>4</sub>) or an [[ammonium]] ion (NH<sub>4</sub><sup>+</sup>), four hydrogen atoms surround a central carbon or nitrogen atom with tetrahedral symmetry. For this reason, one of the leading journals in organic chemistry is called ''[[Tetrahedron (journal)|Tetrahedron]]''. See also [[tetrahedral molecular geometry]]. The [[central angle]] between any two vertices of a perfect tetrahedron is <math>\arccos{\left(-\tfrac{1}{3}\right)}</math>, or approximately 109.47°.

[[Water]], H<sub>2</sub>O, also has a tetrahedral structure, with two hydrogen atoms and two lone pairs of electrons around the central oxygen atoms. Its tetrahedral symmetry is not perfect, however, because the lone pairs repel more than the single O-H bonds.

Quaternary phase diagrams in chemistry are represented graphically as tetrahedra.

However, quaternary phase diagrams in [[communication engineering]] are represented graphically on a two-dimensional plane.

===Electricity and electronics===
{{main|Electricity|Electronics}}
If six equal [[resistor]]s are [[solder]]ed together to form a tetrahedron, then the resistance measured between any two vertices is half that of one resistor.<ref>{{cite journal |last=Klein |first=Douglas J. |year=2002 |title=Resistance-Distance Sum Rules |journal=Croatica Chemica Acta |volume=75 |issue=2 |pages=633–649 |url=http://jagor.srce.hr/ccacaa/CCA-PDF/cca2002/v75-n2/CCA_75_2002_633_649_KLEIN.pdf |format=PDF |accessdate=2006-09-15 }}</ref><ref>Záležák, Tomáš (18 October 2007); [http://ganymed.math.muni.cz/ks/uploads/File/kafe/cj/uloha2.1.5en.pdf "Resistance of a regular tetrahedron"] (PDF), retrieved 25 Jan 2011</ref>

Since [[silicon]] is the most common [[semiconductor]] used in [[solid-state electronics]], and silicon has a [[valence (chemistry)|valence]] of four, the tetrahedral shape of the four chemical bonds in silicon is a strong influence on how [[crystal]]s of silicon form and what shapes they assume.

===Games===
{{main|Game}}
The [[Royal Game of Ur]], dating from 2600 BC, was played with a set of tetrahedral dice.

Especially in [[roleplaying]], this solid is known as a [[4-sided die]], one of the more common [[polyhedral dice]], with the number rolled appearing around the bottom or on the top vertex. Some [[Rubik's Cube]]-like puzzles are tetrahedral, such as the [[Pyraminx]] and [[Pyramorphix]].

The net of a tetrahedron also makes the famous [[Triforce]] from Nintendo's [[The Legend of Zelda]] franchise.

===Color space===
{{main|Color space}}
Tetrahedra are used in color space conversion algorithms specifically for cases in which the luminance axis diagonally segments the color space (e.g. RGB, CMY).<ref>{{cite journal |last=Vondran |first=Gary L. |date=April 1998 |title=Radial and Pruned Tetrahedral Interpolation Techniques |journal=HP Technical Report |volume=HPL-98-95 |pages=1–32 |url=http://www.hpl.hp.com/techreports/98/HPL-98-95.pdf |format=PDF }}</ref>

==={{anchor|Contemporary Art}}Contemporary art===
{{main|Contemporary art}}
The Austrian artist [[Martina Schettina]] created a tetrahedron using [[fluorescent lamp]]s. It was shown at the light art biennale Austria 2010.<ref>[http://lightart.posterous.com/ Lightart-Biennale Austria 2010]</ref>

It is used as album artwork, surrounded by black flames on ''[[The End of All Things to Come]]'' by [[Mudvayne]].

==={{anchor|Popular Culture}}Popular culture===
[[Stanley Kubrick]] originally intended the [[Monolith (Space Odyssey)|monolith]] in ''[[2001: A Space Odyssey (film)|2001: A Space Odyssey]]'' to be a tetrahedron, according to [[Marvin Minsky]], a cognitive scientist and expert on [[artificial intelligence]] who advised Kubrick on the [[Hal 9000]] computer and other aspects of the movie. Kubrick scrapped the idea of using the tetrahedron as a visitor who saw footage of it did not recognize what it was and he did not want anything in the movie regular people did not understand.<ref>{{cite web |title=Marvin Minsky: Stanley Kubrick Scraps the Tetrahedron |url=http://www.webofstories.com/play/53140?o=R |publisher=Web of Stories |accessdate=20 February 2012 }}</ref>

In Season 6, Episode 15 of Futurama, aptly named Möbius Dick, the Planet Express crew pass through an area in space known as the Bermuda Tetrahedron. Where many other ships passing through the area have mysteriously disappeared, including that of the first Planet Express crew.

In the 2013 film ''[[Oblivion (2013 film)|Oblivion]]'' the large structure in orbit above the Earth is of a tetrahedron design and referred to as the Tet.

===Geology===
{{main|Geology}}
The [[tetrahedral hypothesis]], originally published by [[William Lowthian Green]] to explain the formation of the Earth,<ref>{{cite book |title=Vestiges of the Molten Globe, as exhibited in the figure of the earth, volcanic action and physiography |volume=Part I |authorlink=William Lowthian Green |first=William Lowthian |last=Green |publisher=E. Stanford |place=London |year=1875 |oclc=3571917 |url=http://books.google.com/books?id=9DkDAAAAQAAJ }}</ref> was popular through the early 20th century.<ref>{{cite book |authorlink=Arthur Holmes |first=Arthur |last=Holmes |title=Principles of physical geology |url=http://books.google.com/books?id=XUJRAAAAMAAJ |year=1965 |publisher=Nelson |page=32 }}</ref><ref>{{cite news |authorlink=Charles Henry Hitchcock |first=Charles Henry |last=Hitchcock |editor-first=Newton Horace |editor-last=Winchell |title=William Lowthian Green and his Theory of the Evolution of the Earth's Features |work=The American Geologist |url=http://books.google.com/books?id=_Ty8AAAAIAAJ&pg=PA1 |date=January 1900 |volume=XXV |publisher=Geological Publishing Company |pages=1–10 }}</ref>

===Structural engineering===
A tetrahedron having stiff edges is inherently rigid. For this reason it is often used to stiffen frame structures such as [[spaceframe]]s.

===Aviation===
At some [[airfield]]s, a large frame in the shape of a tetrahedron with two sides covered with a thin material is mounted on a rotating pivot and always points into the wind.  Its construction to be big enough to be seen from the air and is sometimes illuminated.  Its purpose is to serve as a reference to pilots indicating wind direction.<ref>{{citation|title=Pilot's Handbook of Aeronautical Knowledge|author=Federal Aviation Administration|publisher=U. S. Government Printing Office|year=2009|isbn=9780160876110|page=13-<!-- This is a hyphen, not a dash. It separates a section number from a page number and does not indicate a page range. -->10|url=http://books.google.com/books?id=0l8WO6Drz50C&pg=SA13-PA10}}.</ref>

==See also==
* [[Boerdijk–Coxeter helix]]
* [[Caltrop]]
* [[Demihypercube]] and [[simplex]] – ''n''-dimensional analogues
* [[Hill tetrahedron]]
* [[Pentachoron]] – 4-dimensional Hyper-Tetrahedron
* [[Schläfli orthoscheme]]
* [[Tetra Pak]]
* [[Tetrahedral kite]]
* [[Tetrahedral number]]
* [[Tetrahedron packing]]
* [[Triangular dipyramid]] – constructed by joining two tetrahedra along one face
* [[Trirectangular tetrahedron]]

==References==
{{Reflist}}

==External links==
{{Commons category |Tetrahedron }}
* {{mathworld |urlname=Tetrahedron |title=Tetrahedron }}
* {{mathworld |urlname=MongesTetrahedronTheorem |title=Monge point }}
* {{mathworld |urlname=EulerPoints |title=Euler points }}
* {{KlitzingPolytopes |polyhedra.htm |3D convex uniform polyhedra |x3o3o - tet }}
* [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
* [http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=csVPXfQQCmqwDYVhk5XGE2m&name=Tetrahedron#applet Editable printable net of a tetrahedron with interactive 3D view]
* [http://polyhedra.org/poly/show/0/tetrahedron Tetrahedron: Interactive Polyhedron Model]
* [http://www.korthalsaltes.com/model.php?name_en=tetrahedron Free paper models of a tetrahedron and many other polyhedra]
* [http://mathforum.org/pcmi/hstp/resources/dodeca/paper.html ''An Amazing, Space Filling, Non-regular Tetrahedron''] that also includes a description of a "rotating ring of tetrahedra", also known as a [[kaleidocycle]].
* [http://www.nyu.edu/its/pubs/connect/spring04/kyriannis_nyunet3.html Tetrahedron Core Network] Application of a tetrahedral structure to create resilient partial-mesh data network
* [http://www.journaldatabase.org/articles/87064/Explicit_Exact_Formulas_f.html Explicit exact formulas for the inertia tensor of an arbitrary tetrahedron in terms of its vertex coordinates]
* {{cite web|title=ViHart rolls a Tetrahedral Dice Yahtzee|url=http://www.numberphile.com/videos/yahtzee_vihart.html|work=Numberphile|publisher=[[Brady Haran]]|author=Hart, Vi|authorlink=Vi Hart}}

{{Polyhedra}}
{{Polyhedron navigator}}
{{Polytopes}}
{{Use dmy dates|date=March 2011}}

[[Category:Deltahedra]]
[[Category:Platonic solids]]
[[Category:Self-dual polyhedra]]
[[Category:Prismatoid polyhedra]]
[[Category:Pyramids and bipyramids]]