Elongated triangular cupola

In geometry, the elongated triangular cupola is a polyhedron constructed from a hexagonal prism by attaching a triangular cupola. It is an example of a Johnson solid.

Construction

The elongated triangular cupola is constructed from a hexagonal prism by attaching a triangular cupola onto one of its bases, a process known as the elongation.Template:R This cupola covers the hexagonal face so that the resulting polyhedron has four equilateral triangles, nine squares, and one regular hexagon.Template:R A convex polyhedron in which all of the faces are regular polygons is the Johnson solid. The elongated triangular cupola is one of them, enumerated as the eighteenth Johnson solid $J_{18}$.Template:R

Properties

The surface area of an elongated triangular cupola $A$ 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 $a$, its surface and volume can be formulated as:Template:R $$\begin{array}{ccc}A& =\frac{18+5\sqrt{3}}{2}{a}^2& \approx 13.330a^2,\\ V& =\frac{5\sqrt{2}+9\sqrt{3}}{6}{a}^3& \approx 3.777a^3.\end{array}$$

File:J18 elongated triangular cupola.stl

3D model of an elongated triangular cupola

It has the three-dimensional same symmetry as the triangular cupola, the cyclic group $C_{3\mathrm{v}}$ of order 6. Its dihedral angle can be calculated by adding the angle of a triangular cupola and a hexagonal prism:Template:R

• 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.

External links

• Template:Mathworld2