Augmented hexagonal prism

The augmented hexagonal prism is constructed by attaching one equilateral square pyramid onto the square face of a hexagonal prism, a process known as augmentation.{{r|rajwade}} This construction involves the removal of the prism square face and replacing it with the square pyramid, so that there are eleven faces: four equilateral triangles, five squares, and two regular hexagons.{{r|berman}} A convex polyhedron in which all of the faces are regular is a Johnson solid, and the augmented hexagonal prism is among them, enumerated as $J\_\{54\}$.{{r|francis}} Relatedly, two or three equilateral square pyramids attaching onto more square faces of the prism give more different Johnson solids; these are the parabiaugmented hexagonal prism $J\_\{55\}$, the metabiaugmented hexagonal prism $J\_\{56\}$, and the triaugmented hexagonal prism $J\_\{57\}$.{{r|rajwade}}

An augmented hexagonal prism with edge length $a$ has surface area{{r|berman}}

$$\backslash left(5\; +\; 4\backslash sqrt\{3\}\backslash right)a^2\; \backslash approx\; 11.928a^2,$$

the sum of two hexagons, four equilateral triangles, and five squares area. Its volume{{r|berman}}

$$\backslash frac\{\backslash sqrt\{2\}\; +\; 9\backslash sqrt\{3\}\}\{2\}a^3\; \backslash approx\; 2.834a^3,$$

can be obtained by slicing into one equilateral square pyramid and one hexagonal prism, and adding their volume up.{{r|berman}}

It has an axis of symmetry passing through the apex of a square pyramid and the centroid of a prism square face, rotated in a half and full-turn angle. Its dihedral angle can be obtained by calculating the angle of a square pyramid and a hexagonal prism in the following:{{r|johnson}}

• The dihedral angle of an augmented hexagonal prism between two adjacent triangles is the dihedral angle of an equilateral square pyramid, $\backslash arccos\; \backslash left(-1/3\backslash right)\; \backslash approx\; 109.5^\backslash circ$
• The dihedral angle of an augmented hexagonal prism between two adjacent squares is the interior of a regular hexagon, $2\backslash pi/3\; =\; 120^\backslash circ$
• The dihedral angle of an augmented hexagonal prism between square-to-hexagon is the dihedral angle of a hexagonal prism between its base and its lateral face, $\backslash pi/2$
• The dihedral angle of a square pyramid between triangle (its lateral face) and square (its base) is $\backslash arctan\; \backslash left(\backslash sqrt\{2\}\backslash right)\; \backslash approx\; 54.75^\backslash circ$. Therefore, the dihedral angle of an augmented hexagonal prism between square-to-triangle and between triangle-to-hexagon, on the edge in which the square pyramid and hexagonal prism are attached, are $$\backslash begin\{align\}$$

\arctan \left(\sqrt{2}\right) + \frac{2\pi}{3} \approx 174.75^\circ, \\

\arctan \left(\sqrt{2}\right) + \frac{\pi}{2} \approx 144.75^\circ.

\end{align}

.

{{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 = 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

}}

{{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 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 = 84–89

| publisher = Hindustan Book Agency

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

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

}}

}}

• {{MathWorld2 | urlname2=JohnsonSolid | title2=Johnson Solid| urlname=AugmentedHexagonalPrism | title=Augmented hexagonal prism }}

Category:Johnson solids

{{Johnson solids navigator}}