Elongated triangular cupola

<h2 id="construction">Construction</h2>
<p>The elongated triangular cupola is constructed from a <a href="/facts/Hexagonal_prism/oxYc0nF2">hexagonal prism</a> by attaching a <a href="/facts/Triangular_cupola/Pyocot8L">triangular cupola</a> onto one of its bases, a process known as the <a href="/facts/Elongation_(geometry)/8SR7CyNz">elongation</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> This cupola covers the hexagonal face so that the resulting polyhedron has four <a href="/facts/Equilateral_triangle/MJomsYDS">equilateral triangles</a>, nine <a href="/facts/Square/JHJeMvmL">squares</a>, and one <a href="/facts/Regular_hexagon/DPzuCzWE">regular hexagon</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> A <a href="/facts/Convex_set/vdAuJRJl">convex</a> polyhedron in which all of the faces are <a href="/facts/Regular_polygon/ws89Nhpt">regular polygons</a> is the <a href="/facts/Johnson_solid/8SR7CyNz">Johnson solid</a>. The elongated triangular cupola is one of them, enumerated as the eighteenth Johnson solid 
  
    
      
        
          J
          
            18
          
        
      
    
    {\displaystyle J_{18}}
  
.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<h2 id="properties">Properties</h2>
<p>The surface area of an elongated triangular cupola 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is the sum of all polygonal face's area. The volume of an elongated triangular cupola can be ascertained by dissecting it into a cupola and a hexagonal prism, after which summing their volume. Given the edge length 
  
    
      
        a
      
    
    {\displaystyle a}
  
, its surface and volume can be formulated as:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>

A
              
              
                
                =
                
                  
                    
                      18
                      +
                      5
                      
                        
                          3
                        
                      
                    
                    2
                  
                
                
                  a
                  
                    2
                  
                
              
              
                ≈
                13.330
                
                  a
                  
                    2
                  
                
                ,
              
            
            
              
                V
              
              
                
                =
                
                  
                    
                      5
                      
                        
                          2
                        
                      
                      +
                      9
                      
                        
                          3
                        
                      
                    
                    6
                  
                
                
                  a
                  
                    3
                  
                
              
              
                ≈
                3.777
                
                  a
                  
                    3
                  
                
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}A&={\frac {18+5{\sqrt {3}}}{2}}a^{2}&\approx 13.330a^{2},\\V&={\frac {5{\sqrt {2}}+9{\sqrt {3}}}{6}}a^{3}&\approx 3.777a^{3}.\end{aligned}}}

</p>

<p>It has the <a href="/facts/Point_groups_in_three_dimensions/Uc4MRIyl">three-dimensional same symmetry</a> as the triangular cupola, the <a href="/facts/Cyclic_group/JL5zfFuL">cyclic group</a> 
  
    
      
        
          C
          
            3
            
              v
            
          
        
      
    
    {\displaystyle C_{3\mathrm {v} }}
  
 of order 6. Its <a href="/facts/Dihedral_angle/C9WEJsuS">dihedral angle</a> can be calculated by adding the angle of a triangular cupola and a hexagonal prism:<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>
<ul><li>the dihedral angle of an elongated triangular cupola between square-to-triangle is that of a triangular cupola between those: 125.3°;</li>
<li>the dihedral angle of an elongated triangular cupola between two adjacent squares is that of a hexagonal prism, the internal angle of its base 120°;</li>
<li>the dihedral angle of a hexagonal prism between square-to-hexagon is 90°, that of a triangular cupola between square-to-hexagon is 54.7°, and that of a triangular cupola between triangle-to-hexagonal is an 70.5°. Therefore, the elongated triangular cupola between square-to-square and triangle-to-square, on the edge where a triangular cupola is attached to a hexagonal prism, is 90° + 54.7° = 144.7° and 90° + 70.5° = 166.5° respectively.</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/ElongatedTriangularCupola.html">Elongated triangular cupola</a>" ("<a href="http://mathworld.wolfram.com/JohnsonSolid.html">Johnson solid</a>") at <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul>

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

<ol>
<li id="fn:1"><p>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></p></li>
<li id="fn:2"><p>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></p></li>
<li id="fn:3"><p>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></p></li>
<li id="fn:4"><p>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:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>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:5" class="footnote-back-ref">↩</a></p></li>
</ol>

Elongated triangular cupola open-in-new

Elongated triangular cupola