geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the elongated pentagonal bipyramid is a polyhedron constructed by attaching two

pentagonal pyramid In geometry, a pentagonal pyramid is a Pyramid (geometry), pyramid with a pentagon base and five triangular faces, having a total of six faces. It is categorized as a Johnson solid if all of the edges are equal in length, forming Equilateral tria ...

s onto the base of a

pentagonal prism In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with seven faces, fifteen edges, and ten vertices. As a semiregular (or uniform) polyhedron If faces are all regular, the pentagonal prism is ...

. It is an example of

Johnson solid In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two Solid geometry, s ...

Construction

The elongated pentagonal bipyramid is constructed from a

by attaching two

s onto its bases, a process called elongation. These pyramids cover the pentagonal faces so that the resulting polyhedron ten

equilateral triangles An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

and five

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

s. A

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

polyhedron in which all of the faces are

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...

s is the

. The elongated pentagonal bipyramid is among them, enumerated as the sixteenth Johnson solid

J_

Properties

The

surface area The surface area (symbol ''A'') of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the d ...

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
 A &= \frac \left(2+\sqrt\right)a^2 \approx 9.330a^2, \\
 V &= \frac \left(5+\sqrt+3 \sqrt\right)a^3 \approx 2.324a^3.
\end

It has the same three-dimensional symmetry group as the pentagonal prism, the dihedral group

D_

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°. * the dihedral angle of an elongated pentagonal bipyramid between two adjacent square faces is that of a regular pentagonal prism, the

internal angle In geometry, an angle of a polygon is formed by two adjacent edge (geometry), sides. For a simple polygon (non-self-intersecting), regardless of whether it is Polygon#Convexity and non-convexity, convex or non-convex, this angle is called an ...

of a

regular pentagon In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...

, 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 In geometry, the pentagonal bifrustum or truncated pentagonal bipyramid is the third in an infinite series of bifrustum polyhedra. It has 10 trapezoidal and 2 pentagonal faces. Constructions The pentagonal bifrustum is the dual polyhedron of a J ...

References

External links

* Johnson solids Bipyramids {{Polyhedron-stub