# Triangular cupola

Triangular cupola
Rank3
TypeCRF
Notation
Bowers style acronymTricu
Coxeter diagramox3xx&#x
Stewart notationQ3
Elements
Faces
Edges3+3+3+6
Vertices3+6
Vertex figures3 rectangles, edge lengths 1 and 2
6 scalene triangles, edge lengths 1, 2, 3
Measures (edge length 1)
Volume${\displaystyle {\frac {5{\sqrt {2}}}{6}}\approx 1.17851}$
Dihedral angles3–4: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
3–6: ${\displaystyle \arccos \left({\frac {1}{3}}\right)\approx 70.52878^{\circ }}$
4–6: ${\displaystyle \arccos \left({\frac {\sqrt {3}}{3}}\right)\approx 54.73561^{\circ }}$
Height${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Central density1
Number of external pieces8
Level of complexity10
Related polytopes
ArmyTricu
RegimentTricu
DualSemibisected trigonal trapezohedron
ConjugateNone
Abstract & topological properties
Flag count60
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryA2×I, order 6
Flag orbits10
ConvexYes
NatureTame

The triangular cupola (OBSA: tricu), is one of the 92 Johnson solids (J3). It consists of 1+3 triangles, 3 squares, and 1 hexagon. It is a cupola based on the equilateral triangle, and is one of three Johnson solid cupolas, the other two being the square cupola and the pentagonal cupola.

It can also be constructed as one of the halves formed by cutting a cuboctahedron in half along a hexagonal section.

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,{\frac {\sqrt {6}}{3}}\right)}$,
• ${\displaystyle \left(0,\,{\frac {\sqrt {3}}{3}},\,{\frac {\sqrt {6}}{3}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,0\right)}$,
• ${\displaystyle \left(\pm 1,\,0,\,0\right)}$.

These coordinates can be formed by putting a triangle and hexagon in parallel planes and setting edge lengths to be equal.

Alternatively, it can be formed by removing the vertices of a triangular face from the cuboctahedron, producing the following coordinates:

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

Multiplying these coordinates by ${\displaystyle {\sqrt {2}}}$, gives integral coordinates for the triangular cupola.

## Representations

A triangular cupola can be represented by the following Coxeter diagrams:

• ox3xx&#xt
• so6ox&#xt

## General variant

The most general variant of the triangular cupola uses a triangle and a ditrigon as bases, connected by 3 isosceles triangles and 3 isosceles trapezoids.

It occurs as the vertex figure of 3 uniform polychora, namely the small ditrigonary icosidodecahedral antiprism, small tritrigonary prismatohecatonicosidishexacosichoron, and great tritrigonary hexacositrishecatonicosachoron. Members of these regiments will have facetings of the triangular cupola for vertex figures.

## Related polyhedra

Two triangular cupolas can be attached at their hexagonal bases in the same orientation to form a triangular orthobicupola. If the second cupola is rotated by 60º the result is the triangular gyrobicupola, better known as the cuboctahedron.

A hexagonal prism can be attached to the triangular cupola's hexagonal base to form the elongated triangular cupola. If a hexagonal antiprism is attached instead, the result is the gyroelongated triangular cupola.