Pentagonal pyramid
TypePyramid
Johnson
J1J2J3
Faces5 triangles
1 pentagon
Edges10
Vertices6
Vertex configuration${\displaystyle 5\times (3^{2}\times 5)+1\times 3^{5))$[1]
Symmetry group${\displaystyle C_{5v))$
Dual polyhedronself-dual
Propertiesconvex, elementary
Net

In geometry, pentagonal pyramid is a pyramid with a pentagon base and five triangular faces, having a total of six faces. If all of the edges are equal in length, the triangular faces are equilateral, and this pyramid is the second Johnson solid ${\displaystyle J_{2))$. The pentagonal pyramid can be found in many polyhedrons, especially in constructing new polyhedra. Its structure can be used in stereochemistry.

## Properties

A pentagonal pyramid has six vertices, ten edges, and six faces. One of its faces is pentagon, a base of the pyramid; five others are triangles.[2] Five of the edges make up the pentagon by connecting its five vertices, and the other five edges are known as the lateral edges of the pyramid, meeting at the sixth vertex called the apex.[3] When all edges are equal in length, the five triangular faces are equilateral and the base is a regular pentagon. This pyramid has the property of Johnson solid ${\displaystyle J_{2))$, a convex polyhedron that all of the faces are regular polygons.[4] The dihedral angle between two adjacent triangular faces and that between the triangular face and the base is ${\displaystyle 138.1897}$ and ${\displaystyle 37.3775}$, respectively.[1]

Given that ${\displaystyle a}$ is the length of all edges of the pentagonal pyramid. A polyhedron's surface area is the sum of the areas of its faces. Therefore, the surface area of a pentagonal pyramid is the sum of the four triangles and one pentagon area. In the case of Johnson solid, the surface area is given the expression:[5] ${\displaystyle A={\frac {a^{2)){2)){\sqrt ((\frac {5}{2))\left(10+{\sqrt {5))+{\sqrt {75+30{\sqrt {5))))\right)))\approx 3.88554a^{2}.}$ The volume ${\displaystyle V}$ of every pyramid equals one-third of the area of its base multiplied by its height. That is, the volume of a pentagonal pyramid is one-third of the product of the height and a pentagonal pyramid's area.[6] Expressed in the same edges length, its volume is:[5] ${\displaystyle V=\left({\frac {5+{\sqrt {5))}{24))\right)a^{3}\approx 0.30150a^{3}.}$

Like other right pyramids with a regular polygon as a base, this pyramid has pyramidal symmetry of cyclic group ${\displaystyle C_{5v))$: the pyramid is left invariant by rotations of one, two, three, and four in five of a full turn around its axis of symmetry, the line connecting the apex to the center of the base. It is also mirror symmetric relative to any perpendicular plane passing through a bisector of the base.[1] It can be represented as the wheel graph ${\displaystyle W_{5))$; more generally, a wheel graph ${\displaystyle W_{n))$ is the representation of the skeleton of a ${\displaystyle n}$-sided pyramid.[7]

The pentagonal pyramid with regular faces is elementary polyhedra. This means it cannot be separated by a plane to create two small convex polyhedrons with regular faces.[8]

## Applications

### In the appearance and construction of polyhedra

Pentakidodecahedron is constructed by augmentation of a dodecahedron
Pentagonal pyramids can be found in a small stellated dodecahedron

Pentagonal pyramids can be found as components of many polyhedrons. Attaching its base to the pentagonal face of another polyhedron is an example of the construction process known as augmentation, and attaching it to prisms or antiprisms is known as elongation or gyroelongation, respectively.[9] An example polyhedron is the pentakis dodecahedron, constructed from the dodecahedron by attaching the base of pentagonal pyramids onto each pentagonal face.[10] The small stellated dodecahedron is a dodecahedron stellated by pentagonal pyramids.[11]

Some Johnson solids are constructed by either augmenting pentagonal pyramids or augmenting other shapes with pentagonal pyramids: elongated pentagonal pyramid ${\displaystyle J_{9))$, gyroelongated pentagonal pyramid ${\displaystyle J_{11))$, pentagonal bipyramid ${\displaystyle J_{13))$, elongated pentagonal bipyramid ${\displaystyle J_{16))$, augmented dodecahedron ${\displaystyle J_{58))$, parabiaugmented dodecahedron ${\displaystyle J_{59))$, metabiaugmented dodecahedron ${\displaystyle J_{60))$, and triaugmented dodecahedron ${\displaystyle J_{61))$.[12] Relatedly, the removal of a pentagonal pyramid from polyhedra is an example known as diminishment; metabidiminished icosahedron ${\displaystyle J_{62))$ and tridiminished icosahedron ${\displaystyle J_{63))$ are the examples that removed pentagonal pyramids from a regular icosahedron.[5]

### Stereochemistry

In stereochemistry, an atom cluster can have a pentagonal pyramidal geometry. This molecule has a main-group element with one active lone pair, which can be described by a model that predicts the geometry of molecules known as VSEPR theory.[13]

## Notes

1. ^ a b c
2. ^
3. ^ Smith (2000), p. 98.
4. ^ Uehara (2020), p. 62.
5. ^ a b c
6. ^
7. ^
8. ^
9. ^
10. ^
11. ^ Kappraff (2001), p. 309.
12. ^ Rajwade (2001), pp. 84–88. See Table 12.3, where ${\displaystyle P_{n))$ denotes the ${\displaystyle n}$-sided prism and ${\displaystyle A_{n))$ denotes the ${\displaystyle n}$-sided antiprism.
13. ^