Augmented hexagonal prism

Augmented hexagonal prism
Type	Johnson J53 – J54 – J55
Faces	4 triangles 5 squares 2 hexagons
Edges	22
Vertices	13
Vertex configuration	2×4(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).

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 ${\displaystyle J_{54}}$.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 ${\displaystyle J_{55}}$, the metabiaugmented hexagonal prism ${\displaystyle J_{56}}$, and the triaugmented hexagonal prism ${\displaystyle J_{57}}$.¹

An augmented hexagonal prism with edge length ${\displaystyle a}$ has surface area² $${\displaystyle \left(5+4{\sqrt {3}}\right)a^{2}\approx 11.928a^{2},}$$ the sum of two hexagons, four equilateral triangles, and five squares area. Its volume² $${\displaystyle {\frac {{\sqrt {2}}+9{\sqrt {3}}}{2}}a^{3}\approx 2.834a^{3},}$$ 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, ${\displaystyle \arccos \left(-1/3\right)\approx 109.5^{\circ }}$
• The dihedral angle of an augmented hexagonal prism between two adjacent squares is the interior of a regular hexagon, ${\displaystyle 2\pi /3=120^{\circ }}$
• 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, ${\displaystyle \pi /2}$
• The dihedral angle of a square pyramid between triangle (its lateral face) and square (its base) is ${\displaystyle \arctan \left({\sqrt {2}}\right)\approx 54.75^{\circ }}$. 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 $${\displaystyle {\begin{aligned}\arctan \left({\sqrt {2}}\right)+{\frac {2\pi }{3}}\approx 174.75^{\circ },\\\arctan \left({\sqrt {2}}\right)+{\frac {\pi }{2}}\approx 144.75^{\circ }.\end{aligned}}}$$.