# Pentagon

Pentagon | |
---|---|

Rank | 2 |

Type | Regular |

Notation | |

Bowers style acronym | Peg |

Coxeter diagram | x5o () |

Schläfli symbol | {5} |

Elements | |

Edges | 5 |

Vertices | 5 |

Vertex figure | Dyad, length (1+√5)/2 |

Measures (edge length 1) | |

Circumradius | |

Inradius | |

Area | |

Angle | 108° |

Central density | 1 |

Number of external pieces | 5 |

Level of complexity | 1 |

Related polytopes | |

Army | Peg |

Dual | Pentagon |

Conjugate | Pentagram |

Abstract & topological properties | |

Flag count | 10 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | H_{2}, order 10 |

Flag orbits | 1 |

Convex | Yes |

Nature | Tame |

The **pentagon** is a polygon with 5 sides. A regular pentagon has equal sides and equal angles.

The only stellation of the pentagon is the pentagram. It and the hexagon are the only polygons with one possible stellation. It and the octagon, decagon, and dodecagon are the only polygons with a single non-compound stellation.

Regular pentagons form the faces of one of the Platonic solids, namely the dodecahedron, along with one of the Kepler–Poinsot solids, namely the great dodecahedron. Pentagons are the highest regular convex polygon to feature in regular polyhedra, and also the highest where a CRF pyramid or cupola is possible.

## Naming[edit | edit source]

The name *pentagon* is derived from the Ancient Greek *πέντε* (5) and *γωνία* (angle), referring to the number of vertices.

Other names include:

**peg**, Bowers style acronym, short for "pentagon"**5-gon**

The combining prefix in BSAs is **pe-**, as in **pe**dip.

## Vertex coordinates[edit | edit source]

Coordinates for the vertices of a regular pentagon of edge length 1, centered at the origin, are:

## Representations[edit | edit source]

A regular pentagon can be represented by the following Coxeter diagrams:

- x5o () (full symmetry)
- ofx&#xt (axial)
- ooooo&#xr (irregular)

## In vertex figures[edit | edit source]

The regular pentagon appears as a vertex figure in two uniform polyhedra, namely the icosahedron (with an edge length of 1) and the small stellated dodecahedron (with an edge length of (√5-1)/2). Irregular pentagons further appear as the vertex figures of some snub polyhedra.

## Other kinds of pentagons[edit | edit source]

The regular pentagon cannot tile the plane by its own without overlap, as the angles around each vertex would not be able to add up to 360°. However, it has been proven that there are exactly 15 families of convex pentagons that can tile the plane.^{[1]}

Various non-regular pentagons exist, all generally having at most a mirror symmetry, or no symmetry at all.

## Stellations[edit | edit source]

The pentagram is the only stellation of the pentagon.

## References[edit | edit source]

- ↑ Rao, Michaël (2017). "Exhaustive search of convex pentagons which tile the plane" (PDF).

## External links[edit | edit source]

- Bowers, Jonathan. "Regular Polygons and Other Two Dimensional Shapes".

- Klitzing, Richard. "Polygons"
- Wikipedia contributors. "Pentagon".
- Hi.gher.Space Wiki Contributors. "Pentagon".

- Hartley, Michael. "{5}*10".