Pentagon

Pentagon
(OFF file)
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	${\sqrt {\frac {5+{\sqrt {5}}}{10}}}\approx 0.85065$
Inradius	${\sqrt {\frac {5+2{\sqrt {5}}}{20}}}\approx 0.68819$
Area	${\frac {\sqrt {25+10{\sqrt {5}}}}{4}}\approx 1.72048$
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	H2, order 10
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

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 pedip.

Vertex coordinates

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

• $\left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}}\right),$
• $\left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right).$

Representations

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

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

In vertex figures

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

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, its has been proven that there are exactly 15 families of convex pentagons that can tile the plane.

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

Stellations

The pentagram is the only stellation of the pentagon.

External links

• 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".