9th Johnson solid (11 faces)
3D model of an elongated pentagonal pyramid In geometry , the elongated pentagonal pyramid is one of the Johnson solids (J 9 pentagonal pyramid (J 2 pentagonal prism to its base.
A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids , Archimedean solids , prisms , or antiprisms ). They were named by Norman Johnson , who first listed these polyhedra in 1966.[ 1]
The following formulae for the height (
H
{\displaystyle H}
surface area (
A
{\displaystyle A}
volume (
V
{\displaystyle V}
L
{\displaystyle L}
[ 2]
H
=
L
⋅ ⋅ -->
(
1
+
5
− − -->
5
10
)
≈ ≈ -->
L
⋅ ⋅ -->
1.525731112
{\displaystyle H=L\cdot \left(1+{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx L\cdot 1.525731112}
A
=
L
2
⋅ ⋅ -->
20
+
5
3
+
25
+
10
5
4
≈ ≈ -->
L
2
⋅ ⋅ -->
8.88554091
{\displaystyle A=L^{2}\cdot {\frac {20+5{\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}}{4}}\approx L^{2}\cdot 8.88554091}
V
=
L
3
⋅ ⋅ -->
(
5
+
5
+
6
25
+
10
5
24
)
≈ ≈ -->
L
3
⋅ ⋅ -->
2.021980233
{\displaystyle V=L^{3}\cdot \left({\frac {5+{\sqrt {5}}+6{\sqrt {25+10{\sqrt {5}}}}}{24}}\right)\approx L^{3}\cdot 2.021980233}
Dual polyhedron
The dual of the elongated pentagonal pyramid has 11 faces: 5 triangular, 1 pentagonal and 5 trapezoidal. It is topologically identical to the Johnson solid.
Dual elongated pentagonal pyramid
Net of dual
See also
References
External links
