Elongated triangular cupola
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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.
ConstructionEdit
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 <math> J_{18} </math>.Template:R
PropertiesEdit
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:Template:R <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>
It has the 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: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.