# Augmented triangular prism

Augmented triangular prism Rank3
TypeCRF
SpaceSpherical
Notation
Bowers style acronymAutip
Coxeter diagramoxx oxo&#xt
Elements
Faces2+2+2 triangles, 2 squares
Edges1+2+2+4+4
Vertices1+2+4
Vertex figures1 square, edge length 1
4 trapezoids, edge lengths 1, 1, 1, 2
2 isosceles triangles, edge lengths 1, 2, 2
Measures (edge length 1)
Volume$\frac{2\sqrt2+3\sqrt3}{12} ≈ 0.66871$ Dihedral angles3–3 join: $\arccos\left(-\frac{\sqrt6}{3}\right) ≈ 144.73561^\circ$ 3–4 join: $\arccos\left(-\frac{3\sqrt2-\sqrt3}{6}\right) ≈ 114.73561^\circ$ 3–3 pyramidal: $\arccos\left(-\frac13\right) ≈ 109.47122^\circ$ 3–4 prismatic: 90°
4–4: 60°
Central density1
Number of external pieces8
Level of complexity13
Related polytopes
ArmyAutip
RegimentAutip
DualLateromonotruncated triangular tegum
ConjugateExcavated triangular prism
Abstract & topological properties
Flag count52
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryK2×I, order 4
ConvexYes
NatureTame

The augmented triangular prism is one of the 92 Johnson solids (J49). It consists of 2+2+2 triangles and 2 squares. It can be constructed by attaching a square pyramid to one of the square faces of the triangular prism.

## Vertex coordinates

An augmented triangular prism of edge length 1 has the following vertices:

• $\left(±\frac12,\,-\frac{\sqrt3}{6},\,±\frac12\right),$ • $\left(0,\,\frac{\sqrt3}{3},\,±\frac12\right),$ • $\left(0,\,-\frac{3\sqrt2+\sqrt3}{6},\,0\right).$ ## In vertex figures

The augmented triangular prism appears as the vertex figure of the nonuniform digonal-triangular duoantiprism. This vertex figure has an edge length of 1, and has no corealmic realization, because the Johnson augmented triangular prism has no circumscribed sphere.

Variants made by changing the edge opposite to the vertex also appear as the vertex figure of the nonuniform duoantiprisms made out of a dyad and a regular polygon.