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Revision as of 14:13, 29 May 2002 by Damian Yerrick (talk | contribs) (Clarified regular polys section)(diff) ← Previous revision | Latest revision (diff) | Newer revision → (diff)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.
Regular polygons have sides that are of equal length and have equal angles between successive pairs of sides.
Concave polygons have at least one internal angle that is greater than 180°, whereas convex polygons have all internal angles less than 180°.
A concyclic or cyclic polygon has all of its vertexes lying on the same circle.
A polygon can belong to several classifications simultaneously.
For example, a square is a regular, convex, cyclic polygon.
Regular Polygons
| Name | Sides | Angle* |
|---|---|---|
| Equilateral triangle | 3 | 60° |
| Square | 4 | 90° |
| Regular pentagon | 5 | 108° |
| Regular hexagon | 6 | 120° |
| Regular heptagon | 7 | 128.57° (approx.) |
| Regular octagon | 8 | 135° |
| Regular nonagon | 9 | 140° |
| Regular decagon | 10 | 144° |
| Regular hectagon | 100 | 176.4° |
| Regular megagon | 10 | 179.99964° |
| Regular googolgon | 10 | 180° (approx.) |
* 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), ..., (xn, yn) 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 + ... + xny1 - x1yn)
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.