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In geometry, the augmented hexagonal prism is one of the Johnson solids (Template:Math). As the name suggests, it can be constructed by augmenting a hexagonal prism by attaching a square pyramid (Template:Math) to one of its equatorial faces. When two or three such pyramids are attached, the result may be a parabiaugmented hexagonal prism (Template:Math), a metabiaugmented hexagonal prism (Template:Math), or a triaugmented hexagonal prism (Template:Math).
ConstructionEdit
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.Template:R 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.Template:R 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 <math> J_{54} </math>.Template:R 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 <math> J_{55} </math>, the metabiaugmented hexagonal prism <math> J_{56} </math>, and the triaugmented hexagonal prism <math> J_{57} </math>.Template:R
PropertiesEdit
An augmented hexagonal prism with edge length <math> a </math> has surface areaTemplate:R <math display="block"> \left(5 + 4\sqrt{3}\right)a^2 \approx 11.928a^2, </math> the sum of two hexagons, four equilateral triangles, and five squares area. Its volumeTemplate:R <math display="block"> \frac{\sqrt{2} + 9\sqrt{3}}{2}a^3 \approx 2.834a^3, </math> can be obtained by slicing into one equilateral square pyramid and one hexagonal prism, and adding their volume up.Template:R
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:Template:R
- The dihedral angle of an augmented hexagonal prism between two adjacent triangles is the dihedral angle of an equilateral square pyramid, <math> \arccos \left(-1/3\right) \approx 109.5^\circ </math>
- The dihedral angle of an augmented hexagonal prism between two adjacent squares is the interior of a regular hexagon, <math> 2\pi/3 = 120^\circ </math>
- 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, <math> \pi/2 </math>
- The dihedral angle of a square pyramid between triangle (its lateral face) and square (its base) is <math> \arctan \left(\sqrt{2}\right) \approx 54.75^\circ </math>. 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 <math display="block"> \begin{align}
\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} </math>.