Elongated pentagonal pyramid

9th Johnson solid (11 faces) From Wikipedia, the free encyclopedia

Elongated pentagonal pyramid

The elongated pentagonal pyramid is a polyhedron constructed by attaching one pentagonal pyramid onto one of the pentagonal prism's bases, a process known as elongation. It is an example of composite polyhedron.[1][2] This construction involves the removal of one pentagonal face and replacing it with the pyramid. The resulting polyhedron has five equilateral triangles, five squares, and one pentagon as its faces.[3] It remains convex, with the faces are all regular polygons, so the elongated pentagonal pyramid is Johnson solid, enumerated as the sixteenth Johnson solid .[4]

Quick Facts Type, Faces ...
Elongated pentagonal pyramid
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TypeJohnson
J8J9J10
Faces5 triangles
5 squares
1 pentagon
Edges20
Vertices11
Vertex configuration5(42.5)
5(32.42)
1(35)
Symmetry groupC5v, [5], (*55)
Rotation groupC5, [5]+, (55)
Dual polyhedronself
Propertiesconvex
Net
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3D model of an elongated pentagonal pyramid

For edge length , an elongated pentagonal pyramid has a surface area by summing the area of all faces, and volume by totaling the volume of a pentagonal pyramid's Johnson solid and regular pentagonal prism:[3]

The elongated pentagonal pyramid has a dihedral between its adjacent faces:[5]

  • the dihedral angle between adjacent squares is the internal angle of the prism's pentagonal base, 108°;
  • the dihedral angle between the pentagon and a square is the right angle, 90°;
  • the dihedral angle between adjacent triangles is that of a regular icosahedron, 138.19°; and
  • 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°.

References

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