The triangular hebesphenorotunda is named by Johnson (1966), 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.^[1] Therefore, the triangular hebesphenorotunda has 20 faces: 13 equilateral triangles, 3 squares, 3 regular pentagons, and 1 regular hexagon.^[2] The faces are all regular polygons, categorizing the triangular hebesphenorotunda as the Johnson solid, enumerated the last one ${\displaystyle J_{92}}$ .^[3] It is elementary polyhedra, meaning that it cannot be separated by a plane into two small regular-faced polyhedra.^[4]

The surface area of a triangular hebesphenorotunda of edge length ${\displaystyle a}$  as:^[2] $${\displaystyle A=\left(3+{\frac {1}{4}}{\sqrt {1308+90{\sqrt {5}}+114{\sqrt {75+30{\sqrt {5}}}}}}\right)a^{2}\approx 16.389a^{2},}$$  and its volume as:^[2] $${\displaystyle V={\frac {1}{6}}\left(15+7{\sqrt {5}}\right)a^{3}\approx 5.10875a^{3}.}$$ 

The triangular hebesphenorotunda with edge length ${\displaystyle {\sqrt {5}}-1}$  can be constructed by the union of the orbits of the Cartesian coordinates: $${\displaystyle {\begin{aligned}\left(0,-{\frac {2}{\tau {\sqrt {3}}}},{\frac {2\tau }{\sqrt {3}}}\right),\qquad &\left(\tau ,{\frac {1}{{\sqrt {3}}\tau ^{2}}},{\frac {2}{\sqrt {3}}}\right)\\\left(\tau ,-{\frac {\tau }{\sqrt {3}}},{\frac {2}{{\sqrt {3}}\tau }}\right),\qquad &\left({\frac {2}{\tau }},0,0\right),\end{aligned}}}$$  under the action of the group generated by rotation by 120° around the z-axis and the reflection about the yz-plane. Here, ${\displaystyle \tau }$  denotes the golden ratio.^[5]

• Weisstein, Eric W., "Triangular hebesphenorotunda" ("Johnson solid") at MathWorld.