Editing Augmented hexagonal prism (section)

== Properties ==
An augmented hexagonal prism with edge length <math> a </math> has surface area{{r|berman}}
<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 volume{{r|berman}}
<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.{{r|berman}}

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:{{r|johnson}}
* 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>.