View source for Elongated triangular pyramid

{{Short description|Polyhedron constructed with tetrahedra and a triangular prism}}
{{Infobox polyhedron
| image          = elongated_triangular_pyramid.png
| type           = [[Johnson solid|Johnson]]<br>{{math|[[pentagonal rotunda|''J''{{sub|6}}]] – '''''J''{{sub|7}}''' – [[elongated square pyramid|''J''{{sub|8}}]]}}
| faces          = 4 [[triangle]]s<br>3 [[Square (geometry)|square]]s
| edges          = 12
| vertices       = 7
| symmetry       = {{math|C{{sub|3v}}, [3], (*33)}}
| rotation_group = {{math|C{{sub|3}}, [3]{{sup|+}}, (33)}}
| vertex_config  = {{math|1(3{{sup|3}})<br>3(3.4{{sup|2}})<br>3(3{{sup|2}}.4{{sup|2}})}}
| dual           = [[self-dual]]{{r|draghicescu}}
| properties     = [[convex set|convex]]
| net            = Elongated Triangular Pyramid Net.svg
}}
[[File:Tetraedro elongado 3D.stl|thumb|Johnson solid {{math|''J''{{sub|7}}}}.]]

In [[geometry]], the '''elongated triangular pyramid''' is one of the [[Johnson solid]]s ({{math|''J''{{sub|7}}}}).  As the name suggests, it can be constructed by elongating a [[tetrahedron]] by attaching a [[triangular prism]] to its base. Like any elongated [[Pyramid (geometry)|pyramid]], the resulting solid is topologically (but not geometrically) self-[[dual polyhedron|dual]].

== Construction ==
The elongated triangular pyramid is constructed from a [[triangular prism]] by attaching [[regular tetrahedron]] onto one of its bases, a process known as [[Elongation (geometry)|elongation]].{{r|rajwade}} The tetrahedron covers an [[equilateral triangle]], replacing it with three other equilateral triangles, so that the resulting polyhedron has four equilateral triangles and three [[Square (geometry)|square]]s as its faces.{{r|berman}} A convex polyhedron in which all of the faces are [[regular polygon]]s is called the [[Johnson solid]], and the elongated triangular pyramid is among them, enumerated as the seventh Johnson solid <math> J_7 </math>.{{r|uehara}}

== Properties ==
An elongated triangular pyramid with edge length <math> a </math> has a height, by adding the height of a regular tetrahedron and a triangular prism:{{r|pye}}
<math display="block"> \left( 1 + \frac{\sqrt{6}}{3}\right)a \approx 1.816a. </math>
Its surface area can be calculated by adding the area of all eight equilateral triangles and three squares:{{r|berman}}
<math display="block"> \left(3+\sqrt{3}\right)a^2 \approx 4.732a^2, </math>
and its volume can be calculated by slicing it into a regular tetrahedron and a prism, adding their volume up:{{r|berman}}:
<math display="block"> \left(\frac{1}{12}\left(\sqrt{2}+3\sqrt{3}\right)\right)a^3 \approx 0.551a^3. </math>

It has the [[Point groups in three dimensions|three-dimensional symmetry group]], the cyclic group <math> C_{3\mathrm{v}} </math> of order 6. Its [[dihedral angle]] can be calculated by adding the angle of the tetrahedron and the triangular prism:{{r|johnson}}
* the dihedral angle of a tetrahedron between two adjacent triangular faces is <math display="inline"> \arccos \left(\frac{1}{3}\right) \approx 70.5^\circ </math>;
* the dihedral angle of the triangular prism between the square to its bases is <math display="inline"> \frac{\pi}{2} = 90^\circ </math>, and the dihedral angle between square-to-triangle, on the edge where tetrahedron and triangular prism are attached, is <math display="inline"> \arccos \left(\frac{1}{3}\right) + \frac{\pi}{2} \approx 160.5^\circ </math>;
* the dihedral angle of the triangular prism between two adjacent square faces is the internal angle of an equilateral triangle <math display="inline"> \frac{\pi}{3} = 60^\circ </math>.

== References ==
{{reflist|refs=

<ref name="berman">{{cite journal
 | 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="draghicescu">{{cite book
 | last = Draghicescu | first = Mircea
 | contribution = Dual Models: One Shape to Make Them All
 | contribution-url = https://archive.bridgesmathart.org/2016/bridges2016
 | editor-first1 = Eva | editor-last1 = Torrence
 | editor-first2 = Bruce | editor-last2 = Torrence
 | editor-first3 = Carlo H. | editor-last3 = Séquin
 | editor-first4 = Douglas | editor-last4 = McKenna
 | editor-first5 = Kristóf | editor-last5 = Fenyvesi
 | editor-first6 = Reza | editor-last6 = Sarhangi
 | title = Bridges Finland: Mathematics, Music, Art, Architecture, Education, Culture
 | url = https://archive.bridgesmathart.org/2016/frontmatter.pdf
 | pages = 635–640
}}</ref>

<ref name="johnson">{{cite journal
 | 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="pye">{{cite journal
 | last = Sapiña | first = R.
 | title = Area and volume of the Johnson solid <math> J_{8} </math>
 | url = https://www.problemasyecuaciones.com/geometria3D/volumen/Johnson/J8/calculadora-area-volumen-formulas.html
 | issn = 2659-9899
 | access-date = 2020-09-09
 | language = es
 | journal = Problemas y Ecuaciones
}}</ref>

<ref name="rajwade">{{cite book
 | 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
}}</ref>

<ref name="uehara">{{cite book
 | last = Uehara | first = Ryuhei
 | year = 2020
 | title = Introduction to Computational Origami: The World of New Computational Geometry
 | url = https://books.google.com/books?id=51juDwAAQBAJ&pg=PA62
 | page = 62
 | publisher = Springer
 | isbn = 978-981-15-4470-5
 | doi = 10.1007/978-981-15-4470-5
 | s2cid = 220150682
}}</ref>

}}


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

{{Johnson solids navigator}}

{{DEFAULTSORT:Elongated Triangular Pyramid}}
[[Category:Johnson solids]]
[[Category:Self-dual polyhedra]]
[[Category:Pyramids (geometry)]]