In geometry, the elongated triangular orthobicupola 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. 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 35th Johnson solid J_{35}.

Properties

An elongated triangular orthobicupola with a given edge length a has a surface area, by adding the area of all regular faces: 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: It has the same three-dimensional symmetry groups as the triangular orthobicupola, the dihedral group 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, and that between its base and square face is. The dihedral angle of a regular triangular cupola between each triangle and the hexagon is approximately 70.5^\circ, that between each square and the hexagon is 54.7^\circ, and that between square and triangle is 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:

Related polyhedra and honeycombs

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