# Elongated triangular cupola

Elongated triangular cupola
Rank3
TypeCRF
Notation
Bowers style acronymEtcu
Coxeter diagramoxx3xxx&#xt
Elements
Faces
Edges3+3+3+3+3+6+6=27
Vertices3+6+6=15
Vertex figures3 rectangles, edge lengths 1 and 2
6 trapezoids, edge lengths 1, 2, 2, 2
6 isosceles triangles, edge lengths 3, 2, 2
Measures (edge length 1)
Volume${\displaystyle {\frac {5{\sqrt {2}}+9{\sqrt {3}}}{6}}\approx 3.77659}$
Dihedral angles3–4 join: ${\displaystyle \arccos \left(-{\frac {2{\sqrt {2}}}{3}}\right)\approx 160.52878^{\circ }}$
4–4 join: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
3–4 cupolaic: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
4–4 prismatic: 120°
4–6: 90°
Central density1
Number of external pieces14
Level of complexity18
Related polytopes
ArmyEtcu
RegimentEtcu
DualHexakis order-6 truncated semibisected trigonal trapezohedron
ConjugateNone
Abstract & topological properties
Flag count108
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryA2×I, order 6
Flag orbits18
ConvexYes
NatureTame

The elongated triangular cupola (OBSA: etcu) is one of the 92 Johnson solids (J18). It consists of 1+3=4 triangles, 3+3+3=9 squares, and 1 hexagon. It can be constructed by attaching a hexagonal prism to the hexagonal base of the triangular cupola.

If a second cupola is attached to the other hexagonal base of the prism in the same orientation, the result is the elongated triangular orthobicupola. If the second cupola is rotated 60º instead, rhe result is the elongated triangular gyrobicupola.

## Vertex coordinates

An elongated triangular cupola of edge length 1 has the following vertices:

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

## Related polytopes

The elongated triangular cupola can be tunnelled with a triangular cupola and a triangular prism to form a quasi-convex Stewart toroid, the tunnelled elongated triangular cupola. This is the only quasi-convex Stewart toroid with the elongated triangular cupola as its convex hull.