# Square cupola

Square cupola
Rank3
TypeCRF
SpaceSpherical
Bowers style acronymSquacu
Coxeter diagramox4xx&#x
Elements
Vertex figures4 isosceles trapezoids, edge lengths 1, 2, 2, 2
8 scalene triangles, edge lengths 1, 2, 2+2
Faces4 triangles, 1+4 squares, 1 octagon
Edges4+4+4+8
Vertices4+8
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{5+2\sqrt2}}{2} ≈ 1.39897}$
Volume${\displaystyle \frac{3+2\sqrt2}{3} ≈ 1.94281}$
Dihedral angles3–4: ${\displaystyle \arccos\left(-\frac{\sqrt6}{3}\right) ≈ 144.73561°}$
4–4: 135°
3–8: ${\displaystyle \arccos\left(\frac{\sqrt3}{3}\right) ≈ 54.73561°}$
4–8: 45°
Height${\displaystyle \frac{\sqrt2}{2} ≈ 0.70711}$
Central density1
Euler characteristic2
Related polytopes
ArmySquacu
RegimentSquacu
DualSemibisected tetragonal trapezohedron
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryB2×I, order 8
ConvexYes
NatureTame

The square cupola, or squacu, is one of the 92 Johnson solids (J4). It consists of 4 triangles, 1+4 squares, and 1 octagon. It is a cupola based on the square, and is one of three Johnson solid cupolas, the other two being the triangular cupola and the pentagonal cupola.

It can be obtained as a segment of the small rhombicuboctahedron, when considered as an elongated square orthobicupola.

## Vertex coordinates

A square cupola of edge length 1 has vertices given by the following coordinates:

• ${\displaystyle \left(±\frac12,\,±\frac12,\,\frac{\sqrt2}{2}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt2}{2},\,±\frac12,\,0\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{1+\sqrt2}{2},\,0\right).}$

These can be obtained from placing a square and octagon in parallel planes.

## Representations

A square cupola has the following Coxeter diagrams:

• ox4xx&#x
• so8ox&#x
• oqxw qowx&#xr (bases in digonal symmetry)

## Related polyhedra

Two square cupolas can be attached at their octagonal bases in the same orientation to form a square orthobicupola. If the second cupola is rotated by 45º the result is the square gyrobicupola.

An octagonal prism can be attached to the square cupola's octagonal base to form the elongated square cupola. If an octagonal antiprism is attached instead, the result is the gyroelongated square cupola.