elongated triangular pyramid

The elongated triangular pyramid is constructed from a triangular prism by attaching regular tetrahedron onto one of its bases, a process known as 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 squares as its faces.{{r|berman}} A convex polyhedron in which all of the faces are regular polygons is called the Johnson solid, and the elongated triangular pyramid is among them, enumerated as the seventh Johnson solid $J\_7$.{{r|uehara}}

An elongated triangular pyramid with edge length $a$ has a height, by adding the height of a regular tetrahedron and a triangular prism:{{r|pye}}

$$\backslash left(\; 1\; +\; \backslash frac\{\backslash sqrt\{6\}\}\{3\}\backslash right)a\; \backslash approx\; 1.816a.$$

Its surface area can be calculated by adding the area of all eight equilateral triangles and three squares:{{r|berman}}

$$\backslash left(3+\backslash sqrt\{3\}\backslash right)a^2\; \backslash approx\; 4.732a^2,$$

and its volume can be calculated by slicing it into a regular tetrahedron and a prism, adding their volume up:{{r|berman}}:

$$\backslash left(\backslash frac\{1\}\{12\}\backslash left(\backslash sqrt\{2\}+3\backslash sqrt\{3\}\backslash right)\backslash right)a^3\; \backslash approx\; 0.551a^3.$$

It has the three-dimensional symmetry group, the cyclic group $C\_\{3\backslash mathrm\{v\}\}$ 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 $\backslash arccos\; \backslash left(\backslash frac\{1\}\{3\}\backslash right)\; \backslash approx\; 70.5^\backslash circ$;
• the dihedral angle of the triangular prism between the square to its bases is $\backslash frac\{\backslash pi\}\{2\}\; =\; 90^\backslash circ$, and the dihedral angle between square-to-triangle, on the edge where tetrahedron and triangular prism are attached, is $\backslash arccos\; \backslash left(\backslash frac\{1\}\{3\}\backslash right)\; +\; \backslash frac\{\backslash pi\}\{2\}\; \backslash approx\; 160.5^\backslash circ$;
• the dihedral angle of the triangular prism between two adjacent square faces is the internal angle of an equilateral triangle $\backslash frac\{\backslash pi\}\{3\}\; =\; 60^\backslash circ$.

{{reflist|refs=

{{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

}}

{{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

}}

{{cite journal

| last = Sapiña | first = R.

| title = Area and volume of the Johnson solid $J\_\{8\}$

| 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

}}

{{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 = 84–89

| isbn = 978-93-86279-06-4

| doi = 10.1007/978-93-86279-06-4

}}

{{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

}}

}}

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

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Category:Johnson solids

Category:Self-dual polyhedra

Category:Pyramids (geometry)