Pentagon

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Regular pentagon
A regular pentagon
Edges and vertices	5
Schläfli symbol	{5}
Coxeter–Dynkin diagram
Symmetry group	Dihedral (D5)
Internal angle (degrees)	108°
Properties	convex, cyclic, equilateral, isogonal, isotoxal

Look up pentagon in Wiktionary, the free dictionary.

This article is about the geometric figure. For the headquarters of the United States Department of Defense, see The Pentagon. For other uses, see Pentagon (disambiguation).

In geometry, a pentagon From the Greek number 5 (pente) is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540°. A pentagram is an example of a self-intersecting pentagon.

Contents [show] 1 Regular pentagons 1.1 Derivation of the area formula 1.2 Derivation of the diagonal length formula 1.3 Chords from the circumscribing circle to the vertices 2 Construction of a regular pentagon 2.1 Euclid’s methods 2.2 Richmond’s method 2.3 Verification 2.4 Alternative method 2.5 Carlyle circles 2.6 Direct method 2.7 Simple method 3 Graphs 4 Pentagons in nature 4.1 Plants 4.2 Animals 5 Pentagons in tiling 6 In-line notes and references 7 See also 8 External links

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[edit] Regular pentagons

A regular pentagon has all sides of equal length and all interior angles are equal measure (108°). It has five lines of reflectional symmetry and it has rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). Its Schläfli symbol is {5}. The chords of a regular pentagon are in golden ratio to its sides.

The area of a regular convex pentagon with side length t is given by

A pentagram or pentangle is a regular star pentagon. Its Schläfli symbol is {5/2}. Its sides form the diagonals of a regular convex pentagon – in this arrangement the sides of the two pentagons are in the golden ratio.

When a regular pentagon is inscribed in a circle with radius R, its edge length t is given by the expression