Elongated pentagonal bipyramid
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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.
ConstructionEdit
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.Template:R 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 <math> J_{16} </math>.Template:R
PropertiesEdit
The surface area of an elongated pentagonal bipyramid <math> A </math> is the sum of all polygonal faces' area: ten equilateral triangles, and five squares. Its volume <math> V </math> 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 <math> a </math>, they can be formulated as:Template:R <math display="block"> \begin{align}
A &= \frac{5}{2} \left(2+\sqrt{3}\right)a^2 \approx 9.330a^2, \\
V &= \frac{1}{12} \left(5+\sqrt{5}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^3 \approx 2.324a^3.
\end{align} </math>
It has the same three-dimensional symmetry group as the pentagonal prism, the dihedral group <math> D_{5\mathrm{h}} </math> of order 20. Its dihedral angle can be calculated by adding the angle of the pentagonal pyramid and pentagonal prism:Template:R
- the dihedral angle of an elongated pentagonal bipyramid between two adjacent triangular faces is that of a pentagonal pyramid between those, 138.19°.
- the dihedral angle of an elongated pentagonal bipyramid between two adjacent square faces is that of a regular pentagonal prism, the internal angle of a regular pentagon, 108°.
- the dihedral angle of an elongated pentagonal bipyramid between square-to-triangle is the sum of the dihedral angle of a pentagonal pyramid between triangle-to-pentagon with that of a pentagonal prism between square-to-pentagon, 37.38° + 90° = 127.38°.
The dual of the elongated square bipyramid is a pentagonal bifrustum.