Editing Triangular cupola (section)

== Properties ==
The triangular cupola has 4 [[Triangle|triangles]], 3 [[Square (geometry)|squares]], and 1 [[hexagon]] as their faces; the hexagon is the base and one of the four triangles is the top. If all of the edges are equal in length, the [[Equilateral triangle|triangles]] and the hexagon becomes [[Regular polygon|regular]].{{r|berman|uehara}} The [[dihedral angle]] between each triangle and the hexagon is approximately 70.5°, that between each square and the hexagon is 54.7°, and that between square and triangle is 125.3°.{{r|johnson}} A [[Convex polytope|convex]] polyhedron in which all of the faces are regular is a [[Johnson solid]], and the triangular cupola is among them, enumerated as the third Johnson solid <math> J_{3} </math>.{{r|uehara}}

Given that <math> a </math> is the edge length of a triangular cupola. Its surface area <math> A </math> can be calculated by adding the area of four equilateral triangles, three squares, and one hexagon:{{r|berman}}
<math display="block"> A = \left(3+\frac{5\sqrt{3}}{2} \right) a^2 \approx 7.33a^2. </math>
Its height <math> h </math> and volume <math> V </math> is:{{r|pye|berman}}
<math display="block"> \begin{align}
 h &= \frac{\sqrt{6}}{3} a\approx 0.82a, \\
 V &= \left(\frac{5}{3\sqrt{2}}\right)a^3 \approx 1.18a^3.
\end{align}
</math>

[[File:J3 triangular cupola.stl|thumb|3D model of a triangular cupola]]
It has an [[axis of symmetry]] passing through the center of its both top and base, which is symmetrical by rotating around it at one- and two-thirds of a full-turn angle. It is also mirror-symmetric relative to any perpendicular plane passing through a bisector of the hexagonal base. Therefore, it has [[pyramidal symmetry]], the [[cyclic group]] <math> C_{3\mathrm{v}} </math> of order 6.{{r|johnson}}