Augmented hexagonal prism

<h2 id="construction">Construction</h2>
The augmented hexagonal prism is constructed by attaching one <a href="/facts/Equilateral_square_pyramid/wDetDohM">equilateral square pyramid</a> onto the square face of a <a href="/facts/Hexagonal_prism/oxYc0nF2">hexagonal prism</a>, a process known as <a href="/facts/Augmentation_(geometry)/8SR7CyNz">augmentation</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> This construction involves the removal of the prism square face and replacing it with the square pyramid, so that there are eleven faces: four <a href="/facts/Equilateral_triangle/MJomsYDS">equilateral triangles</a>, five <a href="/facts/Square_(geometry)/JHJeMvmL">squares</a>, and two regular <a href="/facts/Hexagon/DPzuCzWE">hexagons</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> 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}}
 
.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> Relatedly, two or three equilateral square pyramids attaching onto more square faces of the prism give more different Johnson solids; these are the <a href="/facts/Parabiaugmented_hexagonal_prism/9nKmL9kn">parabiaugmented hexagonal prism</a> 
 
 
 
 
 J
 
 55
 
 
 
 
 {\displaystyle J_{55}}
 
, the <a href="/facts/Metabiaugmented_hexagonal_prism/fFM7FrGD">metabiaugmented hexagonal prism</a> 
 
 
 
 
 J
 
 56
 
 
 
 
 {\displaystyle J_{56}}
 
, and the <a href="/facts/Triaugmented_hexagonal_prism/G2QoLwN2">triaugmented hexagonal prism</a> 
 
 
 
 
 J
 
 57
 
 
 
 
 {\displaystyle J_{57}}
 
.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<h2 id="properties">Properties</h2>
An augmented hexagonal prism with edge length 
 
 
 
 a
 
 
 {\displaystyle a}
 
 has surface area<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

(
          
            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<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

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.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>
It has an <a href="/facts/Axis_of_symmetry/bpvJqcnS">axis of symmetry</a> 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 <a href="/facts/Dihedral_angle/C9WEJsuS">dihedral angle</a> can be obtained by calculating the angle of a square pyramid and a hexagonal prism in the following:<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

<ul><li>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 }}
 
</li>
<li>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 }}
 
</li>
<li>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}
 
</li>
<li>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}}}
 
.</li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a>, "<a href="https://mathworld.wolfram.com/AugmentedHexagonalPrism.html">Augmented hexagonal prism</a>" ("<a href="http://mathworld.wolfram.com/JohnsonSolid.html">Johnson Solid</a>") at <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Rajwade, A. R. (2001). Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem. Texts and Readings in Mathematics. Hindustan Book Agency. p. 84–89. doi:10.1007/978-93-86279-06-4. ISBN 978-93-86279-06-4. <a href="978-93-86279-06-4" target="_blank">978-93-86279-06-4</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Francis, Darryl (August 2013). "Johnson solids & their acronyms". Word Ways. 46 (3): 177. <a href="https://go.gale.com/ps/i.do?id=GALE%7CA340298118" target="_blank">https://go.gale.com/ps/i.do?id=GALE%7CA340298118</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Rajwade, A. R. (2001). Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem. Texts and Readings in Mathematics. Hindustan Book Agency. p. 84–89. doi:10.1007/978-93-86279-06-4. ISBN 978-93-86279-06-4. <a href="978-93-86279-06-4" target="_blank">978-93-86279-06-4</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Johnson, Norman W. (1966). "Convex polyhedra with regular faces". Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/cjm-1966-021-8. MR 0185507. S2CID 122006114. Zbl 0132.14603. <a href="/wiki/Norman_W._Johnson" target="_blank">/wiki/Norman_W._Johnson</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
</ol>

Augmented hexagonal prism open-in-new

Augmented hexagonal prism