# Tetrahedron

Tetrahedron
Rank3
TypeRegular
Notation
Bowers style acronymTet
Coxeter diagramx3o3o ()
Schläfli symbol{3,3}
Tapertopic notation12
Conway notationT
Stewart notationY3
Elements
Faces4 triangles
Edges6
Vertices4
Vertex figureEquilateral triangle, edge length 1
Petrie polygons3 skew squares
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {6}}{4}}\approx 0.61237}$
Edge radius${\displaystyle {\frac {\sqrt {2}}{4}}\approx 0.35355}$
Inradius${\displaystyle {\frac {\sqrt {6}}{12}}\approx 0.20412}$
Volume${\displaystyle {\frac {\sqrt {2}}{12}}\approx 0.11785}$
Dihedral angle${\displaystyle \arccos \left({\frac {1}{3}}\right)\approx 70.52878^{\circ }}$
HeightsPoint atop trig: ${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Dyad atop perp dyad: ${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Central density1
Number of external pieces4
Level of complexity1
Related polytopes
ArmyTet
RegimentTet
DualTetrahedron
Petrie dualPetrial tetrahedron
κ ?Petrial cube
ConjugateNone
Abstract & topological properties
Flag count24
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryA3, order 24
Flag orbits1
ConvexYes
Net count2
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, as well as one of the star regular polychora, the grand 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. Multiplying these coordinates by ${\displaystyle 2{\sqrt {2}}}$ give integral coordinates.

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

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

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

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

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

Multiplying these coordinates by ${\displaystyle {\sqrt {2}}}$ gives another set of integral coordinates.

## 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:

## 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: