Editing Elongated triangular cupola

{{Short description|Polyhedron with triangular cupola and hexagonal prism}}
{{Infobox polyhedron
|image=elongated_triangular_cupola.png
|type=[[Johnson solid|Johnson]]<br>{{math|[[gyroelongated square bipyramid|''J''{{sub|17}}]] – '''''J''{{sub|18}}''' – [[elongated square cupola|''J''{{sub|19}}]]}}
|faces=4 [[triangle]]s<br>9 [[Square (geometry)|square]]s<br>1 [[hexagon]]
|edges=27
|vertices=15
|symmetry={{math|''C''{{sub|3v}}}}
|vertex_config={{math|6(4{{sup|2}}.6)<br>3(3.4.3.4)<br>6(3.4{{sup|3}})}}
|dual=-
|properties=[[Convex polytope|convex]]
|net=Johnson solid 18 net.png
}}

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 (geometry)|elongation]].{{r|rajwade}} This cupola covers the hexagonal face so that the resulting polyhedron has four [[equilateral triangle]]s, nine [[square]]s, and one [[regular hexagon]].{{r|berman}} A [[convex set|convex]] polyhedron in which all of the faces are [[regular polygon]]s is the [[Johnson solid]]. The elongated triangular cupola is one of them, enumerated as the eighteenth Johnson solid <math> J_{18} </math>.{{r|francis}}

== 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.

==References==
{{Reflist|refs=

<ref name="berman">{{citation
 | last = Berman | first = Martin
 | year = 1971
 | title = Regular-faced convex polyhedra
 | journal = Journal of the Franklin Institute
 | volume = 291
 | issue = 5
 | pages = 329–352
 | doi = 10.1016/0016-0032(71)90071-8
 | mr = 290245
}}.</ref>

<ref name=francis>{{citation|first=Darryl|last=Francis|title=Johnson solids & their acronyms|journal=Word Ways|date=August 2013|volume=46|issue=3|page=177|url=https://go.gale.com/ps/i.do?id=GALE%7CA340298118}}.</ref>

<ref name="johnson">{{citation
 | last = Johnson | first = Norman W. | authorlink = Norman W. Johnson
 | year = 1966
 | title = Convex polyhedra with regular faces
 | journal = [[Canadian Journal of Mathematics]]
 | volume = 18
 | pages = 169–200
 | doi = 10.4153/cjm-1966-021-8
 | mr = 0185507
 | s2cid = 122006114
 | zbl = 0132.14603| doi-access = free
}}.</ref>

<ref name="rajwade">{{citation
 | last = Rajwade | first = A. R.
 | title = Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem
 | series = Texts and Readings in Mathematics
 | year = 2001
 | url = https://books.google.com/books?id=afJdDwAAQBAJ&pg=PA84
 | publisher = Hindustan Book Agency
 | page = 84&ndash;89
 | isbn = 978-93-86279-06-4
 | doi = 10.1007/978-93-86279-06-4
| url-access = subscription
 }}.</ref>


}}

==External links==
* {{mathworld2 | urlname2 = JohnsonSolid | title2 = Johnson solid| urlname =ElongatedTriangularCupola| title =Elongated triangular cupola}}

[[Category:Johnson solids]]

{{Johnson solids navigator}}