Misplaced Pages

This is an old revision of this page, as edited by The Ostrich (talk | contribs) at 15:36, 29 May 2002 (Proposed taxonomy.). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

A polygon (from the Greek poly, for "many", and gwnos, for "angle") is a closed planar path composed of a finite number of straight lines. The term polygon sometimes also refers to the interior of the polygon (the open area that this path encloses) or to the union of both.

The taxonomic classification of polygons is somewhat complex, but the following tree may shed some light on the matter:

                                      Polygon
                                     /       \
                                 Simple     Complex
                                /     \
                           Convex     Concave
                            /
                       Regular

Regular polygons have sides that are of equal length and have equal angles between successive pairs of sides.
Convex polygons have no internal angles greater than 180&deg.
Concave polygons have at least one internal angle that is greater than 180&deg.
Simple polygons are described by a single, non-intersecting boundary.
Complex polygons may have intersecting boundaries.

An additional classification, not part of this taxonomy, is a concyclic or cyclic polygon - all the vertices of which lie on a circle. Note that all triangles have this property (circumcircle).

A polygon belonging to any taxonomic class belongs also to all the superclasses of that class.

For example, a square is a regular, convex, simple polygon (it is also cyclic).

Regular Polygons

* Angle = 180° - 360°/Sides

We will assume Euclidean geometry throughout.

Any polygon, regular or irregular, has as many angles as it has sides, and the sum of its angles is equal to (s-2)×180°, where s is the number of its sides.

The area A of a polygon can be computed if the cartesian coordinates (x₁, y₁), (x₂, y₂), ..., (x_n, y_n) of its vertices, listed in order as the area is circulated in counter-clockwise fashion, are known. The formula is

A = 1/2 · (x₁y₂ - x₂y₁ + x₂y₃ - x₃y₂ + ... + x_ny₁ - x₁y_n)

The question of which regular polygons can be constructed with ruler and compass alone was settled by Gauss when he was 19:

A regular polygon with n sides can be constructed with ruler and compass if and only if the odd prime factors of n are distinct prime numbers of the form 2^(2^k)+1. (The only known primes of this type are 3, 5, 17, 257, 65537.)

See also polyhedron, polytope.