# Square cupola

Square cupola
Rank3
TypeCRF
Notation
Bowers style acronymSquacu
Coxeter diagramox4xx&#x
Stewart notationQ4
Elements
Faces
Edges4+4+4+8
Vertices4+8
Vertex figures4 isosceles trapezoids, edge lengths 1, 2, 2, 2
8 scalene triangles, edge lengths 1, 2, 2+2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {5+2{\sqrt {2}}}}{2}}\approx 1.39897}$
Volume${\displaystyle {\frac {3+2{\sqrt {2}}}{3}}\approx 1.94281}$
Dihedral angles3–4: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
4–4: 135°
3–8: ${\displaystyle \arccos \left({\frac {\sqrt {3}}{3}}\right)\approx 54.73561^{\circ }}$
4–8: 45°
Height${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Central density1
Number of external pieces10
Level of complexity10
Related polytopes
ArmySquacu
RegimentSquacu
DualSemibisected tetragonal trapezohedron
Abstract & topological properties
Flag count80
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryB2×I, order 8
Flag orbits10
ConvexYes
NatureTame

The square cupola 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:

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

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.