Editing Elongated square pyramid

{{Short description|Polyhedron with cube and square pyramid}}
{{Infobox polyhedron
 | image = elongated_square_pyramid.png
 | type = [[Johnson solid|Johnson]]<br>{{math|[[elongated triangular pyramid|''J''{{sub|7}}]] – '''''J''{{sub|8}}''' – [[elongated pentagonal pyramid|''J''{{sub|9}}]]}}
 | faces = 4 [[triangle]]s<br>1+4 [[Square (geometry)|square]]s
 | edges = 16
 | vertices = 9
 | symmetry = <math> C_{4v} </math>
 | angle = {{bulletlist
 | triangle-to-triangle: 109.47°
 | square-to-square: 90°
 | triangle-to-square: 144.74°
}}
 | vertex_config = <math> 4 \times (4^3) </math><br><math> 1 \times (3^4) </math><br><math> 4 \times (3^2 \times 4^2) </math>
 | properties = [[convex set|convex]], [[composite polyhedron|composite]]
 | net = Elongated_Square_Pyramid_Net.svg
}}

In [[geometry]], the '''elongated square pyramid''' is a convex polyhedron constructed from a [[Cube (geometry)|cube]] by attaching an [[equilateral square pyramid]] onto one of its faces. It is an example of [[Johnson solid]].

== Construction ==
The elongated square pyramid is a [[Composite polyhedron|composite]], since it can constructed by attaching one [[Equilateral square pyramid|equilateral square pyramid]] onto one of the faces of a [[Cube (geometry)|cube]], a process known as [[Elongation (geometry)|elongation]] of the pyramid.{{r|timofeenko-2010|rajwade}} One square face of each parent body is thus hidden, leaving five squares and four [[equilateral triangle]]s as faces of the composite.{{r|berman}}

A convex polyhedron in which all of its faces are regular is a [[Johnson solid]], and the elongated square bipyramid is one of them, denoted as <math> J_{15} </math>, the fifteenth Johnson solid.{{r|uehara}}

== Properties ==
Given that <math> a </math> is the edge length of an elongated square pyramid. The height of an elongated square pyramid can be calculated by adding the height of an equilateral square pyramid and a cube. The height of a cube is the same as the edge length of a cube's side, and the height of an equilateral square pyramid is <math> (1/\sqrt{2})a </math>. Therefore, the height of an elongated square bipyramid is:{{r|pye}}
<math display="block"> a + \frac{1}{\sqrt{2}}a = \left(1 + \frac{\sqrt{2}}{2}\right)a \approx 1.707a.  </math>
Its surface area can be calculated by adding all the area of four equilateral triangles and four squares:{{r|berman}}
<math display="block"> \left(5 + \sqrt{3}\right)a^2 \approx 6.732a^2. </math>
Its volume is obtained by slicing it into an equilateral square pyramid and a cube, and then adding them:{{r|berman}}
<math display="block"> \left(1 + \frac{\sqrt{2}}{6}\right)a^3 \approx 1.236a^3. </math>

[[File:Pirámide cuadrada elongada.stl|thumb|3D model of a elongated square pyramid.]]
The elongated square pyramid has the same [[Point groups in three dimensions|three-dimensional symmetry group]] as the equilateral square pyramid, the [[cyclic group]] <math> C_{4v} </math> of order eight. Its [[dihedral angle]] can be obtained by adding the angle of an equilateral square pyramid and a cube:{{r|johnson}}
* The dihedral angle of an elongated square bipyramid between two adjacent triangles is the dihedral angle of an equilateral triangle between its lateral faces, <math> \arccos(-1/3) \approx 109.47^\circ </math>,
* The dihedral angle of an elongated square bipyramid between two adjacent squares is the dihedral angle of a cube between those, <math> \pi/2 = 90^\circ </math>,
* The dihedral angle of an equilateral square pyramid between square and triangle is <math> \arctan \left(\sqrt{2}\right) \approx 54.74^\circ </math>. Therefore, the dihedral angle of an elongated square bipyramid between triangle-to-square, on the edge where the equilateral square pyramids attach the cube, is <math display="block"> \arctan\left(\sqrt{2}\right) + \frac{\pi}{2} \approx 144.74^\circ. </math>

== See also==
*[[Elongated square bipyramid]]

==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="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="timofeenko-2010">{{cite journal
 | last = Timofeenko | first = A. V.
 | year = 2010
 | title = Junction of Non-composite Polyhedra
 | journal = St. Petersburg Mathematical Journal
 | volume = 21 | issue = 3 | pages = 483–512
 | doi = 10.1090/S1061-0022-10-01105-2
 | url = https://www.ams.org/journals/spmj/2010-21-03/S1061-0022-10-01105-2/S1061-0022-10-01105-2.pdf
}}</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 = ElongatedSquarePyramid  | title2 = Elongated square pyramid| urlname = JohnsonSolid | title = Johnson solid}}

{{Johnson solids navigator}}

[[Category:Composite polyhedron]]
[[Category:Johnson solids]]
[[Category:Self-dual polyhedra]]
[[Category:Pyramids (geometry)]]