Elongated triangular orthobicupola

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

In geometry, the elongated triangular orthobicupola or cantellated triangular prism is one of the Johnson solids (J₃₅). As the name suggests, it can be constructed by elongating a triangular orthobicupola by inserting a hexagonal prism between its two halves.

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. This construction process known as elongation, giving the resulting polyhedron has 8 equilateral triangles and 12 squares. A convex polyhedron in which all faces are regular is Johnson solid, and the elongated triangular orthobicupola is one among them, enumerated as 36th Johnson solid $J_{36}$

Properties

An elongated triangular orthobicupola with a given edge length $a$ has a surface area, by adding the area of all regular faces: $$\left(12+2{\sqrt {3}}\right)a^{2}\approx 15.464a^{2}.$$ Its volume can be calculated by cutting it off into two triangular cupolae and a hexagonal prism with regular faces, and then adding their volumes up: $$\left({\frac {5{\sqrt {2}}}{3}}+{\frac {3{\sqrt {3}}}{2}}\right)a^{3}\approx 4.955a^{3}.$$

Related polyhedra and honeycombs

The elongated triangular orthobicupola forms space-filling honeycombs with tetrahedra and square pyramids.

External links

• Weisstein, Eric W., "Elongated triangular orthobicupola" ("Johnson solid") at MathWorld.