Elongated pentagonal bipyramid

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.^[2][3] 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 ${\displaystyle J_{16}}$.^[4]

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

It has the same three-dimensional symmetry group as the pentagonal prism, the dihedral group ${\displaystyle D_{5\mathrm {h} }}$ of order 20. Its dihedral angle can be calculated by adding the angle of the pentagonal pyramid and pentagonal prism:^[5]

• 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.