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Tetrahedron
✖
Tetrahedron

✖

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

✖

represents a tetrahedron with edge length l.

✖
Tetrahedron[{θ,ϕ},…]

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

✖
Tetrahedron[{x,y,z},…]

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

✖
Tetrahedron[{p1,p2,p3,p4}]

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

✖
Tetrahedron[{{p1,1,p1,2,p1,3,p1,4},{p2,1,…},…}]

represents a collection of tetrahedra.

Details and Options

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

A standard unit tetrahedron:

In[3]:=3
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-hovl9
Out[3]=3

A styled tetrahedron:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-ii9xg7
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-84fmk
Out[2]=2

Volume and centroid:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-ia7t48
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-t6dwl
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-bgi1zx
Out[3]=3

Scope  (19)Survey of the scope of standard use cases

Graphics  (9)

Specification  (3)

A unit tetrahedron:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-jmbqy5
Out[1]=1

A single tetrahedron:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-ggb1ww
Out[1]=1

Multiple tetrahedrons:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-1c6641
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-ppozsb
Out[2]=2

Styling  (3)

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

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-38df6s
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-78t7nc
Out[2]=2

Apply a Texture to the faces:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-vff8kd
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-5tt6t
Out[2]=2

Assign VertexColors to vertices:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-le5wxu
Out[1]=1

Coordinates  (3)

Specify coordinates by fractions of the plot range:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-u8d4ip
Out[1]=1

Specify scaled offsets from the ordinary coordinates:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-6qq3xm
Out[1]=1

Points can be Dynamic:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-6gl1zz
Out[1]=1

Regions  (10)

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

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-eq5j78
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-f2f0kz
Out[2]=2

Geometric dimension is the dimension of the shape itself:

In[3]:=3
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-ca28v8
Out[3]=3

Membership testing:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-ghs1r
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-tg4hq
Out[2]=2

Get conditions for membership:

In[3]:=3
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-9ge5ph
Out[3]=3

Volume:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-se0twe
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-e06l44
Out[2]=2

Centroid:

In[3]:=3
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-gwq4b4
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-oknxhk
Out[4]=4

Distance from a point:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-oc6hy
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-bruj1e
Out[2]=2

The equidistance contours for a tetrahedron:

In[3]:=3
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-jdsxbg
Out[3]=3

Signed distance from a point:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-cybvpc
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-bm4ed
Out[2]=2

Nearest point in the region:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-btejnv
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-et4yza
Out[2]=2

Nearest points to an enclosing sphere:

In[3]:=3
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-fccj33
In[4]:=4
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-m59d3a
In[5]:=5
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-g70sch
Out[5]=5

A tetrahedron is bounded:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-dym4fu
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-i3tfrr
Out[2]=2

Find its range:

In[3]:=3
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-l3exhn
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-bx1r15
Out[4]=4

Integrate over a tetrahedron region:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-k224q2
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-banwkr
Out[2]=2

Optimize over a tetrahedron region:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-nf9ton
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-hyz4dq
Out[2]=2

Solve equations in a tetrahedron region:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-bximqe
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-cn6ygq
Out[2]=2

Applications  (5)Sample problems that can be solved with this function

The standard tetrahedron is given by points :

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-mp1io
Out[1]=1

A Kuhn tetrahedron is given by points :

In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-i13vsi
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-cm9tq9

Compute its volume:

In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-724d32
Out[2]=2

Visualize it:

In[3]:=3
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-w890ej
Out[3]=3

Create a compound of two regular tetrahedra:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-ztnf2
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-e375n7
In[3]:=3
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-y48i81
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-drujer

Get the faces of the region:

In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-x4rulk

Compare the area of each face:

In[3]:=3
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-bfs82
Out[3]=3

Visualize the region:

In[4]:=4
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-gcu34i
Out[4]=4

A tetrahedron can be subdivided into eight sub-tetrahedra:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-kclu1m
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-bjkhbc
Out[2]=2

This can be done recursively:

In[3]:=3
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-iepen4
In[4]:=4
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-bkylqv
Out[4]=4

Properties & Relations  (8)Properties of the function, and connections to other functions

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

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-8nmuqi
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-06rw2h
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-6fke8b

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

In[4]:=4
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-bousc5
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-u32usv
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-h426c
Out[2]=2

Point index list of tetrahedra vertices:

In[3]:=3
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-mtv2bj
In[4]:=4
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-sos1yg
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-jufwh0
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-ehckv7
In[3]:=3
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-cc9t05
Out[3]=3

Any tetrahedron is an affine transformation of the standard tetrahedron:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-bs4ml0

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

In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-d94h8v
In[3]:=3
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-mqbtqm

Compare original and transformed unit tetrahedron:

In[4]:=4
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-jwiq78
In[5]:=5
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-jk2vlt
Out[5]=5

Tetrahedron is a special case of Simplex:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-i2yu0m
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-xgjvwm
Out[2]=2

ImplicitRegion can represent any Tetrahedron region:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-hw5ae3
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-q64t4f
Out[2]=2

Tetrahedron is the set of convex combinations of its vertices:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-geptbu
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-k94lg5
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-lq6yxn
In[2]:=2
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-uqdh6e
Out[2]=2

Neat Examples  (2)Surprising or curious use cases

Random collection of tetrahedra:

In[5]:=5
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-fbt4h6
Out[5]=5

Sweep a tetrahedron around an axis:

In[1]:=1
✖
https://wolfram.com/xid/08uo1wtnxcxsnhu-6js1gl
Out[1]=1
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).

Text

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).

CMS

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. Wolfram Research. Last Modified 2019. https://reference.wolfram.com/language/ref/Tetrahedron.html.

APA

Wolfram Language. (2014). Tetrahedron. Wolfram Language & System Documentation Center. Retrieved from 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

BibTeX

@misc{reference.wolfram_2025_tetrahedron, author="Wolfram Research", title="{Tetrahedron}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/Tetrahedron.html}", note=[Accessed: 04-May-2025 ]}

✖
@misc{reference.wolfram_2025_tetrahedron, author="Wolfram Research", title="{Tetrahedron}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/Tetrahedron.html}", note=[Accessed: 04-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_tetrahedron, organization={Wolfram Research}, title={Tetrahedron}, year={2019}, url={https://reference.wolfram.com/language/ref/Tetrahedron.html}, note=[Accessed: 04-May-2025 ]}

✖
@online{reference.wolfram_2025_tetrahedron, organization={Wolfram Research}, title={Tetrahedron}, year={2019}, url={https://reference.wolfram.com/language/ref/Tetrahedron.html}, note=[Accessed: 04-May-2025 ]}