Augmented hexagonal prism explained

Type:	Johnson
Faces:	4 triangles 5 squares 2 hexagons
Edges:	22
Vertices:	13
Dual:	monolaterotruncated hexagonal bipyramid
Net:	Johnson solid 54 net.png

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

Construction

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. 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. 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} . 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} .

Properties

An augmented hexagonal prism with edge length

a has surface area$$\backslash left(5\; +\; 4\backslash sqrt\backslash right)a^2\; \backslash approx\; 11.928a^2,$$the sum of two hexagons, four equilateral triangles, and five squares area. Its volume$$\backslash fraca^3\; \backslash approx\; 2.834a^3,$$can be obtained by slicing into one equilateral square pyramid and one hexagonal prism, and adding their volume up.

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:

• 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) ≈ 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) ≈ 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 $$\backslash begin\; \backslash arctan\; \backslash left(\backslash sqrt\backslash right)\; +\; \backslash frac\; \backslash approx\; 174.75^\backslash circ,\; \backslash \backslash \; \backslash arctan\; \backslash left(\backslash sqrt\backslash right)\; +\; \backslash frac\; \backslash approx\; 144.75^\backslash circ.\backslash end$$