Augmented hexagonal prism

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: • 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} \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} . == References ==