# Tetrahedron

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Tetrahedron
Rank3
TypeRegular
SpaceSpherical
Bowers style acronymTet
Info
Coxeter diagramx3o3o ()
Schläfli symbol{3,3}
Tapertopic notation12
SymmetryA3, order 24
ArmyTet
RegimentTet
Elements
Vertex figureEquilateral triangle, edge length 1
Faces4 triangles
Edges6
Vertices4
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt6}{4} ≈ 0.61237}$
Edge radius${\displaystyle \frac{\sqrt2}{4} ≈ 0.35355}$
Inradius${\displaystyle \frac{\sqrt6}{12} ≈ 0.20412}$
Volume${\displaystyle \frac{\sqrt2}{12} ≈ 0.11785}$
Dihedral angle${\displaystyle \arccos\left(\frac13\right) ≈ 70.52878°}$
HeightsPoint atop trig: ${\displaystyle \frac{\sqrt6}{3} ≈ 0.81650}$
Dyad atop perp dyad: ${\displaystyle \frac{\sqrt2}{2} ≈ 0.70711}$
Central density1
Euler characteristic2
Number of pieces4
Level of complexity1
Related polytopes
DualTetrahedron
Petrie dualPetrial tetrahedron
ConjugateTetrahedron
Properties
ConvexYes
OrientableYes
NatureTame

The tetrahedron or tet, also sometimes called the 3-simplex, is the simplest possible non-degenerate polyhedron. The full symmetry version has 4 equilateral triangles as faces, joining 3 to a vertex, and is one of the 5 Platonic solids. It is the 3-dimensional simplex.

It is the uniform digonal antiprism and regular-faced triangular pyramid. Both of these forms are convex segmentohedra.

A regular tetrahedron of edge length 2 can be inscribed in the unit cube. In fact the tetrahedron is the alternated cube, which makes it the 3D demihypercube. The next larger simplex that can be inscribed in a hypercube is the octaexon.

The tetrahedron occurs as cells of three of the six convex regular polychora, namely the pentachoron, hexadecachoron, and hexacosichoron.

## Vertex coordinates

Coordinates for the vertices of a tetrahedron can be given by all even changes of sign of:

• ${\displaystyle \left(\frac{\sqrt{2}}{4},\,\frac{\sqrt{2}}{4},\,\frac{\sqrt{2}}{4}\right).}$

These arise from the fact that a tetrahedron can be constructed as the alternation of the cube.

Alternate coordinates can be derived from those of the triangle, by considering the tetrahedron as a triangular pyramid:

• ${\displaystyle \left(±\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12}\right),}$
• ${\displaystyle \left(0,\,\frac{\sqrt{3}}{3},\,-\frac{\sqrt{6}}{12}\right),}$
• ${\displaystyle \left(0,\,0,\,\frac{\sqrt{6}}{4}\right).}$

These are more complicated, but generalize to simplexes of any dimension.

Simpler coordinates can be given in four dimensions, as all permutations of:

• ${\displaystyle \left(\frac{\sqrt{2}}{2},\,0,\,0,\,0\right).}$

## Representations

A regular tetrahedron can be represented by the following Coxeter diagrams:

## In vertex figures

Tetrahedra in vertex figures
Name Picture Schläfli symbol Edge length
Pentachoron
{3,3,3} ${\displaystyle 1}$
Tesseract
{4,3,3} ${\displaystyle \sqrt{2}}$
Hecatonicosachoron
{5,3,3} ${\displaystyle \frac{1+\sqrt{5}}{2}}$
Great grand stellated hecatonicosachoron
{5/2,3,3} ${\displaystyle \frac{\sqrt{5}-1}{2}}$
Hexagonal tiling honeycomb
{6,3,3}

## Related polyhedra

Two tetrahedra can be attached at a common face to form a triangular tegum, one of the Johnson solids.

A tetrahedron can also be elongated by attaching a triangular prism to one of the faces, forming the elongated triangular pyramid.

A number of uniform polyhedron compounds are composed of tetrahedra:

o3o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Tetrahedron tet {3,3} x3o3o
Truncated tetrahedron tut t{3,3} x3x3o
Tetratetrahedron = Octahedron oct r{3,3} o3x3o
Truncated tetrahedron tut t{3,3} o3x3x
Tetrahedron tet {3,3} o3o3x
Small rhombitetratetrahedron = Cuboctahedron co rr{3,3} x3o3x
Great rhombitetratetrahedron = Truncated octahedron toe tr{3,3} x3x3x
Snub tetrahedron = Icosahedron ike sr{3,3} s3s3s

## Other kinds of tetrahedra

Besides the regular tetrahedron, there are a number of other polyhedra containing four triangular faces. Tetrahedra are generally classified by symmetry. Some of these classes of tetrahedra include: