# Truncated cube

Truncated cube

Type Archimedean solid
Uniform polyhedron
Elements F = 14, E = 36, V = 24 (χ = 2)
Faces by sides 8{3}+6{8}
Schläfli symbols t{4,3}
t0,1{4,3}
Wythoff symbol 2 3 | 4
Coxeter diagram
Symmetry group Oh, BC3, [4,3], (*432), order 48
Rotation group O, [4,3]+, (432), order 24
Dihedral Angle 3-8:125°15'51"
8-8:90°
References U09, C21, W8
Properties Semiregular convex

Colored faces

3.8.8
(Vertex figure)

Triakis octahedron
(dual polyhedron)

Net

In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices.

If the truncated cube has unit edge length, its dual triakis octahedron has edges of lengths 2 and $\scriptstyle {2+\sqrt{2}}$.

## Area and volume

The area A and the volume V of a truncated cube of edge length a are:

$A = 2\left(6+6\sqrt{2}+\sqrt{3}\right)a^2 \approx 32.4346644a^2$
$V = \frac{1}{3}\left(21+14\sqrt{2}\right)a^3 \approx 13.5996633a^3.$

## Orthogonal projections

The truncated cube has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: triangles, and octagons. The last two correspond to the B2 and A2 Coxeter planes.

Orthogonal projections
Centered by Vertex Edge
3-8
Edge
8-8
Face
Octagon
Face
Triangle
Truncated
cube
Triakis
octahedron
Projective
symmetry
[2] [2] [2] [4] [6]

## Cartesian coordinates

The following Cartesian coordinates define the vertices of a truncated hexahedron centered at the origin with edge length 2ξ:

(±ξ, ±1, ±1),
(±1, ±ξ, ±1),
(±1, ±1, ±ξ)

where ξ = $\scriptstyle {\sqrt2 - 1}$

## Vertex arrangement

It shares the vertex arrangement with three nonconvex uniform polyhedra:

 Truncated cube Nonconvex great rhombicuboctahedron Great cubicuboctahedron Great rhombihexahedron

## Related polyhedra

The truncated cube is one of a family of uniform polyhedra related to the cube and regular octahedron.

Uniform octahedral polyhedra
Symmetry: [4,3], (*432) [4,3]+
(432)
[1+,4,3] = [3,3]
(*332)
[3+,4]
(3*2)
{4,3} t{4,3} r{4,3}
r{31,1}
t{3,4}
t{31,1}
{3,4}
{31,1}
rr{4,3}
s2{3,4}
tr{4,3} sr{4,3} h{4,3}
{3,3}
h2{4,3}
t{3,3}
s{4,3}
s{31,1}

=

=

=
=
or
=
or
=

Duals to uniform polyhedra
V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V33 V3.62 V35

This polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry.

Dimensional family of truncated polyhedra and tilings: 3.2n.2n
Symmetry
*n32
[n,3]
Spherical Euclidean Hyperbolic...
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]

*832
[8,3]...

*∞32
[∞,3]

Truncated
figures

3.4.4

3.6.6

3.8.8

3.10.10

3.12.12

3.14.14

3.16.16

3.∞.∞
Coxeter
Schläfli

t{2,3}

t{3,3}

t{4,3}

t{5,3}

t{6,3}

t{7,3}

t{8,3}

t{∞,3}
Uniform dual figures
Triakis
figures

V3.4.4

V3.6.6

V3.8.8

V3.10.10

V3.12.12

V3.14.14

V3.16.16

V3.∞.∞
Coxeter

It is topologically related to a series of polyhedra and tilings with face configuration Vn.6.6.

Dimensional family of truncated polyhedra and tilings: n.8.8
Symmetry
*n42
[n,4]
Spherical Euclidean Hyperbolic...
*242
[2,4]
D4h
*342
[3,4]
Oh
*442
[4,4]
P4m
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Truncated
figures
2.8.8
3.8.8

4.8.8

5.8.8

6.8.8

7.8.8

8.8.8

∞.8.8
Coxeter
Schläfli

t{4,2}

t{4,3}

t{4,4}

t{4,5}

t{4,6}

t{4,7}

t{4,8}

t{4,∞}
Uniform dual figures
n-kis
figures

V2.8.8

V3.8.8

V4.8.8

V5.8.8

V6.8.8

V7.8.8

V8.8.8

V∞.8.8
Coxeter

### Alternated truncation

A cube can be alternately truncated producing tetrahedral symmetry, with six hexagonal faces, and four triangles at the truncated vertices. It is one of a sequence of alternate truncations of polyhedra and tiling.

## Related polytopes

The truncated cube, is second in a sequence of truncated hypercubes:

 ... Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube