Elongated pentagonal pyramid
9th Johnson solid (11 faces) From Wikipedia, the free encyclopedia
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]
Elongated pentagonal pyramid | |
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Type | Johnson J8 – J9 – J10 |
Faces | 5 triangles 5 squares 1 pentagon |
Edges | 20 |
Vertices | 11 |
Vertex configuration | 5(42.5) 5(32.42) 1(35) |
Symmetry group | C5v, [5], (*55) |
Rotation group | C5, [5]+, (55) |
Dual polyhedron | self |
Properties | convex |
Net | |
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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°.
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