Regular pentagon  

A regular pentagon, {5}  
Edges and vertices  5 
Schläfli symbol  {5} 
Coxeter–Dynkin diagram  
Symmetry group  Dihedral (D_{5}) 
Area (with = edge length)  
Internal angle (degrees)  108° 
Template:WiktionaryparTemplate:Two other uses
In geometry, a pentagon is any fivesided polygon. A pentagon may be simple or selfintersecting. The internal angles in a simple pentagon total .
Regular pentagons
The term pentagon is commonly used to mean a regular convex pentagon, where all sides are equal and all interior angles are equal (to 108°). Its Schläfli symbol is {5}. The chords of this pentagon are in golden ratio to its sides.
The area of a regular convex pentagon with side length 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 , its edge length is given by the expression
Construction
A regular pentagon is constructible using a compass and straightedge, either by inscribing one in a given circle or constructing one on a given edge. This process was described by Euclid in his Elements circa 300 BC.
One method to construct a regular pentagon in a given circle is as follows:
An alternative method is this:
 Draw a circle in which to inscribe the pentagon and mark the center point . (This is the green circle in the diagram to the right).
 Choose a point on the circle that will serve as one vertex of the pentagon. Draw a line through and .
 Construct a line perpendicular to the line passing through . Mark its intersection with one side of the circle as the point .
 Construct the point as the midpoint of .
 Draw a circle centered at through the point . Mark its intersection with the line (inside the original circle) as the point .
 Draw a circle centered at through the point . Mark its intersections with the original (green) circle as the points and .
 Draw a circle centered at through the point . Mark its other intersection with the original circle as the point .
 Draw a circle centered at through the point . Mark its other intersection with the original circle as the point .
 Construct the regular pentagon .
After forming a regular convex pentagon, if you join the nonadjacent corners (drawing the diagonals of the pentagon), you obtain a pentagram, with a smaller regular pentagon in the center. Or if you extend the sides until the nonadjacent ones meet, you obtain a larger pentagram.
A simple method of creating a regular pentagon from just a strip of paper is by tying an overhand knot into the strip and carefully flattening the knot by pulling the ends of the paper strip. Folding one of the ends back over the pentagon will reveal a pentagram when backlit.
Graphs
The complete graph is often drawn as a regular pentagon with all 10 edges connected. This graph also represents an orthographic projection of the 5 vertices and 10 edges of the 5cell. The rectified 5cell, with vertices at the midedges of the 5cell is projected inside a pentagon.
5cell (4D)  Rectified 5cell (4D) 
Pentagons in nature
Plants
Animals
See also
 The Pentagon, headquarters for the U.S. Department of Defense
 Dodecahedron, a polyhedron whose regular form is composed of 12 pentagonal faces
 Trigonometric constants for a pentagon
 Pentagonal numbers
 Associahedron  A pentagon is an order4 associahedron
 Pentagram
 Pentastar, the Chrysler logo
External links
 Weisstein, Eric W., "Pentagon" from MathWorld.
 How to construct a regular pentagon with only a only compass and straightedge.
 How to fold a regular pentagon using only a strip of paper
 Definition and properties of the pentagon, with interactive animation
 Nine constructions for the regular pentagon by Robin Hu
 Renaissance artists' approximate constructions of regular pentagons at Convergence
