Elongated pentagonal pyramid

The elongated pentagonal pyramid is a polyhedron constructed by attaching one <a href="/facts/Pentagonal_pyramid/tN5CBPjK">pentagonal pyramid</a> onto one of the pentagonal prism's bases, a process known as <a href="/facts/Elongation_(geometry)/8SR7CyNz">elongation</a>. It is an example of <a href="/facts/Composite_polyhedron/z1pUdj7r">composite polyhedron</a>. This construction involves the removal of one pentagonal face and replacing it with the pyramid. The resulting polyhedron has five <a href="/facts/Equilateral_triangle/MJomsYDS">equilateral triangles</a>, five squares, and one pentagon as its faces. It remains <a href="/facts/Convex_set/vdAuJRJl">convex</a>, with the faces are all <a href="/facts/Regular_polygon/ws89Nhpt">regular polygons</a>, so the elongated pentagonal pyramid is <a href="/facts/Johnson_solid/8SR7CyNz">Johnson solid</a>, enumerated as the sixteenth Johnson solid 
 
 
 
 
 J
 
 16
 
 
 
 
 {\displaystyle J_{16}}
 
.
For edge length 
 
 
 
 ℓ
 
 
 {\displaystyle \ell }
 
, an elongated pentagonal pyramid has a surface area 
 
 
 
 A
 
 
 {\displaystyle A}
 
 by summing the area of all faces, and volume 
 
 
 
 V
 
 
 {\displaystyle V}
 
 by totaling the volume of a pentagonal pyramid's Johnson solid and regular pentagonal prism:

A
              
              
                
                =
                
                  
                    
                      20
                      +
                      5
                      
                        
                          3
                        
                      
                      +
                      
                        
                          25
                          +
                          10
                          
                            
                              5
                            
                          
                        
                      
                    
                    4
                  
                
                
                  ℓ
                  
                    2
                  
                
                ≈
                8.886
                
                  ℓ
                  
                    2
                  
                
                ,
              
            
            
              
                V
              
              
                
                =
                
                  
                    
                      5
                      +
                      
                        
                          5
                        
                      
                      +
                      6
                      
                        
                          25
                          +
                          10
                          
                            
                              5
                            
                          
                        
                      
                    
                    24
                  
                
                
                  ℓ
                  
                    3
                  
                
                ≈
                2.022
                
                  ℓ
                  
                    3
                  
                
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}A&={\frac {20+5{\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}}{4}}\ell ^{2}\approx 8.886\ell ^{2},\\V&={\frac {5+{\sqrt {5}}+6{\sqrt {25+10{\sqrt {5}}}}}{24}}\ell ^{3}\approx 2.022\ell ^{3}.\end{aligned}}}

The elongated pentagonal pyramid has a dihedral between its adjacent faces:

<ul><li>the dihedral angle between adjacent squares is the <a href="/facts/Internal_angle/jd78Qkbt">internal angle</a> of the prism's pentagonal base, 108°;</li>
<li>the dihedral angle between the pentagon and a square is the right angle, 90°;</li>
<li>the dihedral angle between adjacent triangles is that of a regular icosahedron, 138.19°; and</li>
<li>the dihedral angle between a triangle and an adjacent square is the sum of the angle between those in a pentagonal pyramid and the angle between the base of and the lateral face of a prism, 127.37°.</li></ul>

Elongated pentagonal pyramid open-in-new

Elongated pentagonal pyramid