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. 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 $$ J_{16} $$.

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: $$ \begin{align} A &= \frac{5}{2} \left(2+\sqrt{3}\right)a^2 \approx 9.330a^2, \\ V &= \frac{1}{12} \left(5+\sqrt{5}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^3 \approx 2.324a^3. \end{align} $$

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:
 * the dihedral angle of an elongated pentagonal bipyramid between two adjacent triangular faces is that of a pentagonal pyramid between those, 138.19&deg;.
 * 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&deg;.
 * 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&deg; + 90&deg; = 127.38&deg;.

The dual of the elongated square bipyramid is a pentagonal bifrustum.