Augmented hexagonal prism

TypeJohnson

J 53 - J 54 - J 55

Faces4 triangles
5 squares
2 hexagons

Edges22

Vertices13

Augmented hexagonal prism

Type	Johnson $J 53 - J 54 - J 55$
Faces	4 triangles 5 squares 2 hexagons
Edges	22
Vertices	13
Vertex configuration	$2\times4(4 2 .6) 1(3 4) 4(3 2 .4.6)$
Symmetry group	$C 2v$
Properties	convex
Net

In geometry, the augmented hexagonal prism is one of the Johnson solids ( $J 54$ ). As the name suggests, it can be constructed by augmenting a hexagonal prism by attaching a square pyramid ( $J 1$ ) to one of its equatorial faces. When two or three such pyramids are attached, the result may be a parabiaugmented hexagonal prism ( $J 55$ ), a metabiaugmented hexagonal prism ( $J 56$ ), or a triaugmented hexagonal prism ( $J 57$ ).

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.^[1] 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.^[2] 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 $J_{54}$ .^[3] 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 $J_{57}$ .^[1]

Properties

An augmented hexagonal prism with edge length $a$ has surface area^[2] $\left(5+4{\sqrt {3}}\right)a^{2}\approx 11.928a^{2},$ the sum of two hexagons, four equilateral triangles, and five squares area. Its volume^[2] ${\frac {{\sqrt {2}}+9{\sqrt {3}}}{2}}a^{3}\approx 2.834a^{3},$ can be obtained by slicing into one equilateral square pyramid and one hexagonal prism, and adding their volume up.^[2]

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:^[4]

The dihedral angle of an augmented hexagonal prism between two adjacent triangles is the dihedral angle of an equilateral square pyramid, $\arccos \left(-1/3\right)\approx 109.5^{\circ }$
The dihedral angle of an augmented hexagonal prism between two adjacent squares is the interior of a regular hexagon, $2\pi /3=120^{\circ }$
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, $\pi /2$
The dihedral angle of a square pyramid between triangle (its lateral face) and square (its base) is $\arctan \left({\sqrt {2}}\right)\approx 54.75^{\circ }$ . 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 ${\begin{aligned}\arctan \left({\sqrt {2}}\right)+{\frac {2\pi }{3}}\approx 174.75^{\circ },\\\arctan \left({\sqrt {2}}\right)+{\frac {\pi }{2}}\approx 144.75^{\circ }.\end{aligned}}$ .

v t e Johnson solids
Pyramids, cupolae and rotundae	square pyramid pentagonal pyramid triangular cupola square cupola pentagonal cupola pentagonal rotunda
Modified pyramids	elongated triangular pyramid elongated square pyramid elongated pentagonal pyramid gyroelongated square pyramid gyroelongated pentagonal pyramid triangular bipyramid pentagonal bipyramid elongated triangular bipyramid elongated square bipyramid elongated pentagonal bipyramid gyroelongated square bipyramid
Modified cupolae and rotundae	elongated triangular cupola elongated square cupola elongated pentagonal cupola elongated pentagonal rotunda gyroelongated triangular cupola gyroelongated square cupola gyroelongated pentagonal cupola gyroelongated pentagonal rotunda gyrobifastigium triangular orthobicupola square orthobicupola square gyrobicupola pentagonal orthobicupola pentagonal gyrobicupola pentagonal orthocupolarotunda pentagonal gyrocupolarotunda pentagonal orthobirotunda elongated triangular orthobicupola elongated triangular gyrobicupola elongated square gyrobicupola elongated pentagonal orthobicupola elongated pentagonal gyrobicupola elongated pentagonal orthocupolarotunda elongated pentagonal gyrocupolarotunda elongated pentagonal orthobirotunda elongated pentagonal gyrobirotunda gyroelongated triangular bicupola gyroelongated square bicupola gyroelongated pentagonal bicupola gyroelongated pentagonal cupolarotunda gyroelongated pentagonal birotunda
Augmented prisms	augmented triangular prism biaugmented triangular prism triaugmented triangular prism augmented pentagonal prism biaugmented pentagonal prism augmented hexagonal prism parabiaugmented hexagonal prism metabiaugmented hexagonal prism triaugmented hexagonal prism
Modified Platonic solids	augmented dodecahedron parabiaugmented dodecahedron metabiaugmented dodecahedron triaugmented dodecahedron metabidiminished icosahedron tridiminished icosahedron augmented tridiminished icosahedron
Modified Archimedean solids	augmented truncated tetrahedron augmented truncated cube biaugmented truncated cube augmented truncated dodecahedron parabiaugmented truncated dodecahedron metabiaugmented truncated dodecahedron triaugmented truncated dodecahedron gyrate rhombicosidodecahedron parabigyrate rhombicosidodecahedron metabigyrate rhombicosidodecahedron trigyrate rhombicosidodecahedron diminished rhombicosidodecahedron paragyrate diminished rhombicosidodecahedron metagyrate diminished rhombicosidodecahedron bigyrate diminished rhombicosidodecahedron parabidiminished rhombicosidodecahedron metabidiminished rhombicosidodecahedron gyrate bidiminished rhombicosidodecahedron tridiminished rhombicosidodecahedron
Other elementary solids	snub disphenoid snub square antiprism sphenocorona augmented sphenocorona sphenomegacorona hebesphenomegacorona disphenocingulum bilunabirotunda triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)

Augmented hexagonal prism

Properties

References

External links

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