Pentagon

Pentagon
Rank2
TypeRegular
SpaceSpherical
Bowers style acronymPeg
Info
Coxeter diagramx5o
Schläfli symbol{5}
SymmetryH2, order 10
ArmyPeg
Elements
Edges5
Vertices5
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{5+\sqrt5}{10}} ≈ 0.85065}$
Inradius${\displaystyle \sqrt{\frac{5+2\sqrt5}{20}} ≈ 0.68819}$
Area${\displaystyle \frac{\sqrt{25+10\sqrt5}}{4} ≈ 1.72048}$
Angle108°
Central density1
Euler characteristic0
Number of pieces5
Level of complexity1
Related polytopes
DualPentagon
ConjugatePentagram
Properties
ConvexYes
OrientableYes
NatureTame

The pentagon, or peg, is a polygon with 5 sides. A regular pentagon has equal sides and equal angles.

The combining prefix is pe-, as in pedip.

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 heighst where a CRF pyramid or cupola is possible.

Vertex coordinates

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

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

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.[1]

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.

References

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