Elongated triangular pyramid

<h2 id="construction">Construction</h2>
<p>The elongated triangular pyramid is constructed from a <a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prism</a> by attaching <a href="/facts/Regular_tetrahedron/7mkSrl6j">regular tetrahedron</a> onto one of its bases, a process known as <a href="/facts/Elongation_(geometry)/8SR7CyNz">elongation</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> The tetrahedron covers an <a href="/facts/Equilateral_triangle/MJomsYDS">equilateral triangle</a>, replacing it with three other equilateral triangles, so that the resulting polyhedron has four equilateral triangles and three <a href="/facts/Square_(geometry)/JHJeMvmL">squares</a> as its faces.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> A convex polyhedron in which all of the faces are <a href="/facts/Regular_polygon/ws89Nhpt">regular polygons</a> is called the <a href="/facts/Johnson_solid/8SR7CyNz">Johnson solid</a>, and the elongated triangular pyramid is among them, enumerated as the seventh Johnson solid 
  
    
      
        
          J
          
            7
          
        
      
    
    {\displaystyle J_{7}}
  
.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<h2 id="properties">Properties</h2>
<p>An elongated triangular pyramid with edge length 
  
    
      
        a
      
    
    {\displaystyle a}
  
 has a height, by adding the height of a regular tetrahedron and a triangular prism:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>

(
          
            1
            +
            
              
                
                  6
                
                3
              
            
          
          )
        
        a
        ≈
        1.816
        a
        .
      
    
    {\displaystyle \left(1+{\frac {\sqrt {6}}{3}}\right)a\approx 1.816a.}

Its surface area can be calculated by adding the area of all eight equilateral triangles and three squares:<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>

(
          
            3
            +
            
              
                3
              
            
          
          )
        
        
          a
          
            2
          
        
        ≈
        4.732
        
          a
          
            2
          
        
        ,
      
    
    {\displaystyle \left(3+{\sqrt {3}}\right)a^{2}\approx 4.732a^{2},}

and its volume can be calculated by slicing it into a regular tetrahedron and a prism, adding their volume up:<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>:

(
          
            
              
                1
                12
              
            
            
              (
              
                
                  
                    2
                  
                
                +
                3
                
                  
                    3
                  
                
              
              )
            
          
          )
        
        
          a
          
            3
          
        
        ≈
        0.551
        
          a
          
            3
          
        
        .
      
    
    {\displaystyle \left({\frac {1}{12}}\left({\sqrt {2}}+3{\sqrt {3}}\right)\right)a^{3}\approx 0.551a^{3}.}

</p><p>It has the <a href="/facts/Point_groups_in_three_dimensions/Uc4MRIyl">three-dimensional symmetry group</a>, the cyclic group 
  
    
      
        
          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 the tetrahedron and the triangular prism:<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>
</p>
<ul><li>the dihedral angle of a tetrahedron between two adjacent triangular faces is 
  
    
      
        arccos
        ⁡
        
          (
          
            
              1
              3
            
          
          )
        
        ≈
        
          70.5
          
            ∘
          
        
      
    
    {\textstyle \arccos \left({\frac {1}{3}}\right)\approx 70.5^{\circ }}
  
;</li>
<li>the dihedral angle of the triangular prism between the square to its bases is 
  
    
      
        
          
            π
            2
          
        
        =
        
          90
          
            ∘
          
        
      
    
    {\textstyle {\frac {\pi }{2}}=90^{\circ }}
  
, and the dihedral angle between square-to-triangle, on the edge where tetrahedron and triangular prism are attached, is 
  
    
      
        arccos
        ⁡
        
          (
          
            
              1
              3
            
          
          )
        
        +
        
          
            π
            2
          
        
        ≈
        
          160.5
          
            ∘
          
        
      
    
    {\textstyle \arccos \left({\frac {1}{3}}\right)+{\frac {\pi }{2}}\approx 160.5^{\circ }}
  
;</li>
<li>the dihedral angle of the triangular prism between two adjacent square faces is the internal angle of an equilateral triangle 
  
    
      
        
          
            π
            3
          
        
        =
        
          60
          
            ∘
          
        
      
    
    {\textstyle {\frac {\pi }{3}}=60^{\circ }}
  
.</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/JohnsonSolid.html">Johnson solid</a>" ("<a href="http://mathworld.wolfram.com/ElongatedTriangularPyramid.html">Elongated triangular pyramid</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>Uehara, Ryuhei (2020). Introduction to Computational Origami: The World of New Computational Geometry. Springer. p. 62. doi:10.1007/978-981-15-4470-5. ISBN 978-981-15-4470-5. S2CID 220150682. <a href="978-981-15-4470-5" target="_blank">978-981-15-4470-5</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Sapiña, R. "Area and volume of the Johnson solid 
  
    
      
        
          J
          
            8
          
        
      
    
    {\displaystyle J_{8}}
  
". Problemas y Ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-09-09. <a href="https://www.problemasyecuaciones.com/geometria3D/volumen/Johnson/J8/calculadora-area-volumen-formulas.html" target="_blank">https://www.problemasyecuaciones.com/geometria3D/volumen/Johnson/J8/calculadora-area-volumen-formulas.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><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:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><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:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><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:7" class="footnote-back-ref">↩</a></p></li>
</ol>

Elongated triangular pyramid open-in-new

Elongated triangular pyramid