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16th Johnson solid; pentagonal prism capped by pyramids From Wikipedia, the free encyclopedia
In geometry, the elongated pentagonal bipyramid is a polyhedron constructed by attaching two pentagonal pyramids onto the base of a pentagonal prism. It is an example of Johnson solid.
Elongated pentagonal bipyramid | |
---|---|
Type | Johnson J15 – J16 – J17 |
Faces | 10 triangles 5 squares |
Edges | 25 |
Vertices | 12 |
Vertex configuration | 10(32.42) 2(35) |
Symmetry group | D5h, [5,2], (*522) |
Rotation group | D5, [5,2]+, (522) |
Dual polyhedron | Pentagonal bifrustum |
Properties | convex |
Net | |
The elongated pentagonal bipyramid is constructed from a pentagonal prism by attaching two pentagonal pyramids onto its bases, a process called elongation. These pyramids cover the pentagonal faces so that the resulting polyhedron ten equilateral triangles and five squares.[1][2] A convex polyhedron in which all of the faces are regular polygons is the Johnson solid. The elongated pentagonal bipyramid is among them, enumerated as the sixteenth Johnson solid .[3]
The surface area of an elongated pentagonal bipyramid is the sum of all polygonal faces' area: ten equilateral triangles, and five squares. Its volume can be ascertained by dissecting it into two pentagonal pyramids and one regular pentagonal prism and then adding its volume. Given an elongated pentagonal bipyramid with edge length , they can be formulated as:[2]
It has the same three-dimensional symmetry group as the pentagonal prism, the dihedral group of order 20. Its dihedral angle can be calculated by adding the angle of the pentagonal pyramid and pentagonal prism:[4]
The dual of the elongated square bipyramid is a pentagonal bifrustum.
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