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Polygon names  

Name  Sides  
triangle  3  
quadrilateral  4  
pentagon  5  
hexagon  6  
heptagon[?]  7  
octagon[?]  8  
nonagon[?]  9  
decagon[?]  10  
hectagon[?]  100  
megagon[?]  10^{6}  
googolgon  10^{100} 
The taxonomic classification of polygons is illustrated by the following tree:
Polygon
/ \
Simple Complex
/ \
Convex Concave
/
Regular
A concyclic or cyclic polygon is a polygon whose vertices all lie on a single circle.
For example, a square is a regular, cyclic quadrilateral.
We will assume Euclidean geometry throughout.
Any polygon, regular or irregular, complex or simple, has as many angles as it has sides. Any simple ngon can be considered to be made up of (n2) triangles. The sum of the inner angles of a simple ngon is therfore (n2)π radians (or (n2)180°), and the inner angle of a regular ngon is (n2)π/n radians (or (n2)180°/n).
All regular polygons are concyclic, as are all triangles and equalangled (90°) quadrilaterals(see circumcircle).
The area A of a simple polygon can be computed if the cartesian coordinates (x_{1}, y_{1}), (x_{2}, y_{2}), ..., (x_{n}, y_{n}) of its vertices, listed in order as the area is circulated in counterclockwise fashion, are known. The formula is
If any two simple polygons of equal area are given, then the first can be cut into polygonal pieces which can be reassembled to form the second polygon. This is the BolyaiGerwien theorem.
The question of which regular polygons can be constructed with ruler and compass alone was settled by Carl Friedrich Gauss when he was 19: A regular ngon can be constructed with ruler and compass if and only if the odd prime factors of n are distinct prime numbers of the form
These prime numbers are the Fermat primes; the only known ones are 3, 5, 17, 257 and 65537.
See also: geometric shape, polyhedron, polytope, cyclic polygon.
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