Regular heptagon  

A regular heptagon  
Edges and vertices  7 
Schläfli symbol  {7} 
Coxeter–Dynkin diagram  
Symmetry group  Dihedral (D_{7}) 
Area (with = edge length) 

Internal angle (degrees) 
128.5714286° 
In geometry, a heptagon is a polygon with seven sides and seven angles. In a regular heptagon, in which all sides and all angles are equal, the sides meet at an angle of radians, 128.5714286 degrees. Its Schläfli symbol is {7}. The area of a regular heptagon of side length is given by
The heptagon is also sometimes referred to as the septagon, using "sept" (an elision of septua, a Latinderived numerical prefix, rather than hepta, a Greekderived numerical prefix). The OED lists "septagon" as meaning "heptagonal".
Construction
A regular heptagon is not constructible with compass and straightedge but is constructible with a marked ruler and compass. This type of construction is called a Neusis construction. It is also constructible with compass, straightedge and angle trisector. The impossibility of straightedge and compass construction follows from the observation that is a zero of the irreducible cubic
Consequently this polynomial is the minimal polynomial of , whereas the degree of the minimal polynomial for a constructible number must be a power of 2.
Approximation
A decent approximation for practical use with an accuracy of 0.2% is shown in the drawing. Let A lie on the circumference of the circumcircle. Draw arc BOC. Then gives an approximation for the edge of the heptagon.
Heptagrams
Two kinds of heptagrams can be constructed from regular heptagons, labeled by Schläfli symbols {7/2}, and {7/3}, with the divisor being the interval of connection.
Blue, {7/2} and green {7/3} heptagrams inside a red heptagon.
Uses
The United Kingdom currently (2008) has two heptagonal coins, the 50p and 20p pieces, and the Barbados Dollar is also heptagonal. The 20 eurocent coin has cavities placed similarly. Strictly, the shape of the coins is a curvilinear heptagon to make them curves of constant width: the sides are curved outwards so that the coin will roll smoothly in vending machines. The Brazilian 25 cents coin has a heptagon inscribed in the coin's disk.
Graphs
The K_{7} complete graph is often drawn as a regular heptagon with all 21 edges connected. This graph also represents an orthographic projection of the 7 vertices and 21 edges of the 6simplex. The 21 and 35 vertices of the rectified and birectified 6simplex also orthogonally project into regular heptagons.
6simplex (6D) 
Rectified 6simplex (6D) 
Birectified 6simplex (6D) 
See also
External links
 Definition and properties of a heptagon With interactive animation
 Approximate construction method
 Weisstein, Eric W., "Heptagon" from MathWorld.
 Another approximate construction method
 Polygons  Heptagons
 Recently discovered and highly accurate approximation for the construction of a regular heptagon.

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