Augmented hexagonal prism

Short description: 54th Johnson solid (11 faces)

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
Properties	convex
Net

In geometry, the augmented hexagonal prism is one of the Johnson solids (J₅₄). As the name suggests, it can be constructed by augmenting a hexagonal prism by attaching a square pyramid (J₁) to one of its equatorial faces. When two or three such pyramids are attached, the result may be a parabiaugmented hexagonal prism (J₅₅), a metabiaugmented hexagonal prism (J₅₆), or a triaugmented hexagonal prism (J₅₇).

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

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, $\arccos (-1/3)\approx 109.5^{\circ }$
• 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 $$\begin{array}{r}\arctan \left(\sqrt{2}\right)+\frac{2\pi }{3}\approx 174.75^{\circ },\\ \arctan \left(\sqrt{2}\right)+\frac{\pi }{2}\approx 144.75^{\circ }.\end{array}$$.

External links

• Eric W. Weisstein, Augmented hexagonal prism (Johnson Solid) at MathWorld.