In geometry, the triangular hebesphenorotunda is a Johnson solid with 13 equilateral triangles, 3 squares, 3 regular pentagons, and 1 regular hexagon, making the total of its faces is 20.

Properties

The triangular hebesphenorotunda is named by, with the prefix hebespheno- referring to a blunt wedge-like complex formed by three adjacent lunes—a figure where two equilateral triangles are attached at the opposite sides of a square. The suffix (triangular) -rotunda refers to the complex of three equilateral triangles and three regular pentagons surrounding another equilateral triangle, which bears a structural resemblance to the pentagonal rotunda. Therefore, the triangular hebesphenorotunda has 20 faces: 13 equilateral triangles, 3 squares, 3 regular pentagons, and 1 regular hexagon. The faces are all regular polygons, categorizing the triangular hebesphenorotunda as the Johnson solid, enumerated the last one J_{92}. It is elementary polyhedra, meaning that it cannot be separated by a plane into two small regular-faced polyhedra. The surface area of a triangular hebesphenorotunda of edge length a as: and its volume as:

Cartesian coordinates

The triangular hebesphenorotunda with edge length can be constructed by the union of the orbits of the Cartesian coordinates: under the action of the group generated by rotation by 120° around the z-axis and the reflection about the yz-plane. Here, \tau denotes the golden ratio.