Elongated pentagonal bipyramid

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.

Construction

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 $J_{16}$.Template:R

Properties

The surface area of an elongated pentagonal bipyramid $A$ is the sum of all polygonal faces' area: ten equilateral triangles, and five squares. Its volume $V$ 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 $a$, they can be formulated as:Template:R $$\begin{array}{cc}A& =\frac{5}{2}(2+\sqrt{3})a^2\approx 9.330a^2,\\ V& =\frac{1}{12}(5+\sqrt{5}+3\sqrt{5(5+2\sqrt{5})})a^3\approx 2.324a^3.\end{array}$$

File:J16 elongated pentagonal bipyramid.stl It has the same three-dimensional symmetry group as the pentagonal prism, the dihedral group $D_{5\mathrm{h}}$ 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.

External links

• Template:Mathworld2