Editing Elongated triangular cupola (section)

== Properties ==
The surface area of an elongated triangular cupola <math> A </math> is the sum of all polygonal face's area. The volume of an elongated triangular cupola can be ascertained by dissecting it into a cupola and a hexagonal prism, after which summing their volume. Given the edge length <math> a </math>, its surface and volume can be formulated as:{{r|berman}}
<math display="block">
\begin{align}
 A &= \frac{18 + 5\sqrt{3}}{2}a^2 &\approx 13.330a^2, \\
 V &= \frac{5\sqrt{2} + 9\sqrt{3}}{6}a^3 &\approx 3.777a^3.
\end{align}
</math>

[[File:J18 elongated triangular cupola.stl|thumb|3D model of an elongated triangular cupola]]
It has the [[Point groups in three dimensions|three-dimensional same symmetry]] as the triangular cupola, the [[cyclic group]] <math> C_{3\mathrm{v}} </math> of order 6. Its [[dihedral angle]] can be calculated by adding the angle of a triangular cupola and a hexagonal prism:{{r|johnson}}
* the dihedral angle of an elongated triangular cupola between square-to-triangle is that of a triangular cupola between those: 125.3°;
* the dihedral angle of an elongated triangular cupola between two adjacent squares is that of a hexagonal prism, the internal angle of its base 120°;
* the dihedral angle of a hexagonal prism between square-to-hexagon is 90°, that of a triangular cupola between square-to-hexagon is 54.7°, and that of a triangular cupola between triangle-to-hexagonal is an 70.5°. Therefore, the elongated triangular cupola between square-to-square and triangle-to-square, on the edge where a triangular cupola is attached to a hexagonal prism, is 90° + 54.7° = 144.7° and 90° + 70.5° = 166.5° respectively.