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 the sum of two hexagons, four equilateral triangles, and five squares area. Its volume 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: