represents a regular tetrahedron centered at the origin with unit edge length.


represents a tetrahedron with edge length l.


represents a tetrahedron rotated by an angle θ with respect to the z axis and angle ϕ with respect to the y axis.


represents a tetrahedron centered at {x,y,z}.


represents a general filled tetrahedron with corners p1, p2, p3 and p4.


represents a collection of tetrahedra.

Details and Options


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Basic Examples  (3)

A standard unit tetrahedron:

A styled tetrahedron:

Volume and centroid:

Scope  (19)

Graphics  (9)

Specification  (3)

A unit tetrahedron:

A single tetrahedron:

Multiple tetrahedrons:

Styling  (3)

FaceForm and EdgeForm can be used to specify the styles of the faces and edges:

Apply a Texture to the faces:

Assign VertexColors to vertices:

Coordinates  (3)

Specify coordinates by fractions of the plot range:

Specify scaled offsets from the ordinary coordinates:

Points can be Dynamic:

Regions  (10)

Embedding dimension is the dimension of the space in which the tetrahedron lives:

Geometric dimension is the dimension of the shape itself:

Membership testing:

Get conditions for membership:



Distance from a point:

The equidistance contours for a tetrahedron:

Signed distance from a point:

Nearest point in the region:

Nearest points to an enclosing sphere:

A tetrahedron is bounded:

Find its range:

Integrate over a tetrahedron region:

Optimize over a tetrahedron region:

Solve equations in a tetrahedron region:

Applications  (5)

The standard tetrahedron is given by points :

A Kuhn tetrahedron is given by points :

Define a regular tetrahedron by a radius from its center to a corner:

Compute its volume:

Visualize it:

Create a compound of two regular tetrahedra:

If the four faces of a tetrahedron have the same area, then it is an isosceles tetrahedron:

Get the faces of the region:

Compare the area of each face:

Visualize the region:

A tetrahedron can be subdivided into eight sub-tetrahedra:

This can be done recursively:

Properties & Relations  (8)

TriangulateMesh can be used to decompose a volume mesh into tetrahedra:

Use options such as MaxCellMeasure to control the number of tetrahedra:

A hexahedron can be represented as the union of five tetrahedra:

Point index list of tetrahedra vertices:

A hexahedron can also be represented as the union of six tetrahedra:

Any tetrahedron is an affine transformation of the standard tetrahedron:

The transformation is given by , where A=TemplateBox[{{{, {{{p, _, 1}, -, {p, _, 0}}, ,, ..., ,, {{p, _, 3}, -, {p, _, 0}}}, }}}, Transpose]:

Compare original and transformed unit tetrahedron:

Tetrahedron is a special case of Simplex:

ImplicitRegion can represent any Tetrahedron region:

Tetrahedron is the set of convex combinations of its vertices:

Vertices of a Tetrahedron can be used to form an enclosing Circumsphere:

Neat Examples  (2)

Random collection of tetrahedra:

Sweep a tetrahedron around an axis:

Wolfram Research (2014), Tetrahedron, Wolfram Language function, https://reference.wolfram.com/language/ref/Tetrahedron.html (updated 2019).


Wolfram Research (2014), Tetrahedron, Wolfram Language function, https://reference.wolfram.com/language/ref/Tetrahedron.html (updated 2019).


Wolfram Language. 2014. "Tetrahedron." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2019. https://reference.wolfram.com/language/ref/Tetrahedron.html.


Wolfram Language. (2014). Tetrahedron. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Tetrahedron.html


@misc{reference.wolfram_2024_tetrahedron, author="Wolfram Research", title="{Tetrahedron}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/Tetrahedron.html}", note=[Accessed: 20-July-2024 ]}


@online{reference.wolfram_2024_tetrahedron, organization={Wolfram Research}, title={Tetrahedron}, year={2019}, url={https://reference.wolfram.com/language/ref/Tetrahedron.html}, note=[Accessed: 20-July-2024 ]}