# Augmented cube

Augmented cube Rank3
TypeIsohedral
Elements
Faces24 triangles
Edges12+24
Vertices6+8
Vertex figure8 triambi, 6 squares
Measures (edge length 1)
Circumradius8 (6-fold): ${\frac {\sqrt {3}}{2}}\approx 0.8660254038$ 6 (4-fold): ${\frac {1+{\sqrt {2}}}{2}}\approx 1.207106781$ Volume$1+{\sqrt {2}}\approx 2.414213562$ Number of external pieces24
Related polytopes
DualSemi-uniform hypertruncated octahedron
ConjugateExcavated cube
Convex hullTriakis octahedron
Convex coreSemi-uniform-dual tetrakis cube
Abstract & topological properties
OrientableYes
Properties
SymmetryB3, order 48
ConvexNo
NatureTame

The augmented cube is an isohedral deltahedron. It can be formed by augmenting every face of the regular cube with a square pyramid. It is also a non-convex variant of the tetrakis cube.

An interesting property of this polyhedron is that its convex hull is the exact uniform dual triakis octahedron.

## Vertex coordinates

Coordinates for the vertices of a cube of edge length 1, centered at the origin, are given by all sign changes and permutations of:

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)$ ,
• $\left(\pm {\frac {1+{\sqrt {2}}}{2}},\,0,\,0\right)$ .