Augmented hexagonal prism

Augmented hexagonal prism
Augmented hexagonal prism
Type	Johnson; J53 – J54 – J55
Faces	4 triangles; 5 squares; 2 hexagons
Edges	22
Vertices	13
Vertex configuration	2x4(42.6); 1(34); 4(32.4.6)
Symmetry group	C2v
Dual polyhedron	monolaterotruncated hexagonal bipyramid
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 ( $J 57$ ).

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.^[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 ${\begin{aligned}\arctan \left({\sqrt {2}}\right)+{\frac {2\pi }{3}}\approx 174.75^{\circ },\\\arctan \left({\sqrt {2}}\right)+{\frac {\pi }{2}}\approx 144.75^{\circ }.\end{aligned}}$ .