# Elongated square pyramid

Elongated square pyramid
Rank3
TypeCRF
Notation
Bowers style acronymEsquipy
Coxeter diagramoxx4ooo&#xt
Elements
Faces
Edges4+4+4+4
Vertices1+4+4
Vertex figure1 square, edge length 1; 4 kites, edge lengths 1 and 2; 4 triangles, edge lengths 2
Measures (edge length 1)
Volume${\displaystyle {\frac {6+{\sqrt {2}}}{6}}\approx 1.23570}$
Dihedral angles3–4: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
3–3: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
4–4: 90°
Central density1
Number of external pieces9
Level of complexity8
Related polytopes
ArmyEsquipy
RegimentEsquipy
DualElongated square pyramid
ConjugateRetroelongated square pyramid
Abstract & topological properties
Flag count64
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryB2×I, order 8
Flag orbits8
ConvexYes
NatureTame

The elongated square pyramid (OBSA: esquipy), sometimes called an augmented cube, is one of the 92 Johnson solids (J8). It consists of 4 triangles and 1+4 squares. It can be constructed by attaching a cube, seen as a square prism, to the base of the square pyramid.

If a second pyramid is attached to the opposite face of the cube, the result is the elongated square bipyramid.

## Vertex coordinates

An elongated square pyramid of edge length 1 has the following vertices:

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