Augmented triangular prism: Difference between revisions

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In [[geometry]], the '''augmented triangular prism''' is one of the [[Johnson solid]]s ({{math|''J''{{sub|49}}}}). As the name suggests, it can be constructed by augmenting a triangular [[prism (geometry)|prism]] by attaching a [[square pyramid]] ({{math|''J''{{sub|1}}}}) to one of its equatorial faces. The resulting solid bears a superficial resemblance to the [[gyrobifastigium]] ({{math|''J''{{sub|26}}}}), the difference being that the latter is constructed by attaching a second triangular prism, rather than a square pyramid.

== Construction ==

{{Johnson solid}}

The augmented triangular prism can be constructed from a [[triangular prism]] by attaching an [[equilateral square pyramid]] to one of its square faces.{{r|rajwade}} This square pyramid covers the square face of the prism, so the resulting polyhedron has 6 [[equilateral triangle]]s and 2 [[Square (geometry)|square]]s as its faces.{{r|berman}} A [[Convex set|convex]] polyhedron in which all faces are [[Regular polygon|regular]] is [[Johnson solid]], and the augmented triangular prism is among them, enumerated as 49th Johnson solid <math> J_{49} </math>.{{r|francis}}

== Properties ==

An augmented triangular prism with edge length <math> a </math> has a surface area, calculated by adding six equilateral triangles and two squares' area:{{r|berman}}

<math display="block"> \frac{4 + 3\sqrt{3}}{2}a^2 \approx 4.598a^2. </math>

Its volume can be obtained by slicing it into a regular triangular prism and an equilateral square pyramid, and adding their volume subsequently:{{r|berman}}

<math display="block"> \frac{2\sqrt{2} + 3\sqrt{3}}{12}a^3 \approx 0.669a^3. </math>

It has [[Point groups in three dimensions|three-dimensional symmetry group]] of the cyclic group <math> C_{2\mathrm{v}} </math> of order 4. Its [[dihedral angle]] can be calculated by adding the angle of an equilateral square pyramid and a regular triangular prism. The dihedral angle of an equilateral square pyramid between two adjacent triangular faces is <math display="inline"> \arccos \left(-1/3 \right) \approx 109.5^\circ </math>, and that between a triangular face and its base is <math display="inline"> \arctan \left(\sqrt{2}\right) \approx 54.7^\circ </math>. The dihedral angle of a triangular prism between two adjacent square faces is the [[internal angle]] of an equilateral triangle <math display="inline"> \pi/3 = 60^\circ </math>, and that between square-to-triangle is <math display="inline"> \pi/2 = 90^\circ </math>. Therefore, the dihedral angle of the augmented triangular prism between square-to-triangle and triangle-to-triangle on the edge where both square pyramid and triangular prism are attached is, respectively:{{r|johnson}}

<math display="block"> \begin{align}

\frac{\pi}{3} + \arccos \left(-\frac{1}{3}\right) &\approx 104.5^\circ, \\

\frac{\pi}{2} + \arccos \left(-\frac{1}{3}\right) &\approx 144.5^\circ.

\end{align} </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="francis">{{cite journal

| last = Francis | first = Darryl

| 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">{{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="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

| page = 84&ndash;89

| publisher = Hindustan Book Agency

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

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

}}</ref>

}}

==External links==

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[[Category:Johnson solids]]

{{Polyhedron-stub}}