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).

3D model of an augmented hexagonal prism

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

Properties

An augmented hexagonal prism with edge length a {\displaystyle a} has surface area ( 5 + 4 3 ) a 2 ≈ 11.928 a 2 , {\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 volume 2 + 9 3 2 a 3 ≈ 2.834 a 3 , {\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.

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 ⁡ ( − 1 / 3 ) ≈ 109.5 ∘ {\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, 2 π / 3 = 120 ∘ {\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, π / 2 {\displaystyle \pi /2}
  • The dihedral angle of a square pyramid between triangle (its lateral face) and square (its base) is arctan ⁡ ( 2 ) ≈ 54.75 ∘ {\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 arctan ⁡ ( 2 ) + 2 π 3 ≈ 174.75 ∘ , arctan ⁡ ( 2 ) + π 2 ≈ 144.75 ∘ . {\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}}}.

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