Augmented hexagonal prism

In geometry, the augmented hexagonal prism is one of the Johnson solids ($J53 – J54 – J55$). As the name suggests, it can be constructed by augmenting a hexagonal prism by attaching a square pyramid ($C2v$) to one of its equatorial faces. When two or three such pyramids are attached, the result may be a parabiaugmented hexagonal prism ($2x4(42.6) 1(34) 4(32.4.6)$), a metabiaugmented hexagonal prism ($J54$), or a triaugmented hexagonal prism ($J1$).

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 $$ \left(5 + 4\sqrt{3}\right)a^2 \approx 11.928a^2, $$ the sum of two hexagons, four equilateral triangles, and five squares area. Its volume $$ \frac{\sqrt{2} + 9\sqrt{3}}{2}a^3 \approx 2.834a^3, $$ 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: \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{align} $$.
 * The dihedral angle of an augmented hexagonal prism between two adjacent triangles is the dihedral angle of an equilateral square pyramid, $$ \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{align}