# Gyroelongated square pyramid

Gyroelongated square pyramid Rank3
TypeCRF
SpaceSpherical
Notation
Bowers style acronymGyesp
Coxeter diagramoxo4oox&#xt
Elements
Faces4+4+4 triangles, 1 square
Edges4+4+4+8
Vertices1+4+4
Vertex figures1 square, edge length 1
4 pentagons, edge length 1
4 trapezoids, edge lengths 1, 1, 1, 2
Measures (edge length 1)
Volume$\frac{\sqrt2+2\sqrt{4+3\sqrt2}}{6} ≈ 1.19270$ Dihedral angles3–3 join: $\arccos\left(\frac{1-\sqrt2-2\sqrt{2}}{3}\right) ≈ 158.57177°$ 3–3 antiprismatic: $\arccos\left(\frac{2\sqrt2-1}{3}\right) ≈ 127.55160°$ 3–3 pyramidal: $\arccos\left(-\frac13\right) ≈ 109.47122°$ 3–4: $\arccos\left(-\frac{\sqrt6-\sqrt3}{3}\right) ≈ 103.83616°$ Central density1
Related polytopes
ArmyGyesp
RegimentGyesp
DualOrder-4 monotruncated tetragonal trapezohedron
ConjugateGyroelongated square pyramid
Abstract & topological properties
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryB2×I, order 8
ConvexYes
NatureTame

The gyroelongated square pyramid is one of the 92 Johnson solids (J10). It consists of 4+4+4 triangles and 1 square. It can be constructed by attaching a square antiprism to the base of the square pyramid.

If a second pyramid is attached to the other base of the square antiprism, the result is the gyroelongated square bipyramid.

## Vertex coordinates

A gyroelongated square pyramid of edge length 1 has the following vertices:

• $\left(0,\,0,\,\frac{2\sqrt2+\sqrt{8}}{4}\right),$ • $\left(±\frac12,\,±\frac12,\,\frac{\sqrt{8}}{4}\right),$ • $\left(0,\,±\frac{\sqrt2}{2},\,-\frac{\sqrt{8}}{4}\right),$ • $\left(±\frac{\sqrt2}{2},\,0,\,-\frac{\sqrt{8}}{4}\right).$ 