Elongated triangular orthobicupola
Type	Johnson J34 – J35 – J36
Faces	8 triangles 12 squares
Edges	36
Vertices	18
Vertex configuration	${\displaystyle {\begin{aligned}&6\times (3\times 4\times 3\times 4)+\\&12\times (3\times 4^{3})\end{aligned))}$
Symmetry group	${\displaystyle D_{3h))$
Properties	convex
Net

In geometry, the elongated triangular orthobicupola or cantellated triangular prism^{[citation needed]} is a polyhedron constructed by attaching two regular triangular cupola into the base of a regular hexagonal prism. It is an example of Johnson solid.

Construction

The elongated triangular orthobicupola can be constructed from a hexagonal prism by attaching two regular triangular cupolae onto its base, covering its hexagonal faces.^[1] This construction process known as elongation, giving the resulting polyhedron has 8 equilateral triangles and 12 squares.^[2] A convex polyhedron in which all faces are regular is Johnson solid, and the elongated triangular orthobicupola is one among them, enumerated as 35th Johnson solid ${\displaystyle J_{35))$.^[3]

Properties

An elongated triangular orthobicupola with a given edge length ${\displaystyle a}$ has a surface area, by adding the area of all regular faces:^[2]

$${\displaystyle \left(12+2{\sqrt {3))\right)a^{2}\approx 15.464a^{2}.}$$

$${\displaystyle \left({\frac {5{\sqrt {2))}{3))+{\frac {3{\sqrt {3))}{2))\right)a^{3}\approx 4.955a^{3}.}$$

It has the same three-dimensional symmetry groups as the triangular orthobicupola, the dihedral group ${\displaystyle D_{3h))$ of order 12. Its dihedral angle can be calculated by adding the angle of the triangular cupola and hexagonal prism. The dihedral angle of a hexagonal prism between two adjacent squares is the internal angle of a regular hexagon ${\displaystyle 120^{\circ }=2\pi /3}$, and that between its base and square face is ${\displaystyle \pi /2=90^{\circ ))$. The dihedral angle of a regular triangular cupola between each triangle and the hexagon is approximately ${\displaystyle 70.5^{\circ ))$, that between each square and the hexagon is ${\displaystyle 54.7^{\circ ))$, and that between square and triangle is ${\displaystyle 125.3^{\circ ))$. The dihedral angle of an elongated triangular orthobicupola between the triangle-to-square and square-to-square, on the edge where the triangular cupola and the prism is attached, is respectively:^[4]

$${\displaystyle {\begin{aligned}{\frac {\pi }{2))+70.5^{\circ }&\approx 160.5^{\circ },\\{\frac {\pi }{2))+54.7^{\circ }&\approx 144.7^{\circ }.\end{aligned))}$$

Related polyhedra and honeycombs

The elongated triangular orthobicupola forms space-filling honeycombs with tetrahedra and square pyramids.^[5]