| Revision as of 10:39, 13 October 2002 view sourceThe Anome (talk | contribs)Edit filter managers, Administrators253,491 edits ruler and compass← Previous edit | Revision as of 10:39, 13 October 2002 view source The Anome (talk | contribs)Edit filter managers, Administrators253,491 edits moved link upNext edit → | ||
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| This same formula can also be used to calculate the ''signed area'' of complex polygons: follow the sequence of points and count area to the left of your path positive, to the right negative. | This same formula can also be used to calculate the ''signed area'' of complex polygons: follow the sequence of points and count area to the left of your path positive, to the right negative. | ||
| The question of which regular polygons can be constructed with ruler and compass alone was settled by ] when he was 19: | The question of which regular polygons can be constructed with ] alone was settled by ] when he was 19: | ||
| A regular polygon with ''n'' sides can be constructed with |
A regular polygon with ''n'' sides can be constructed with ruler and compass if and only if the odd ] 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.) | ||
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Revision as of 10:39, 13 October 2002
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 illustrated by the following tree:
Polygon
/ \
Simple Complex
/ \
Convex Concave
/
Regular
- A polygon is simple if it is described by a single, non-intersecting boundary; otherwise it is called complex.
- A simple polygon is called convex if it has no internal angles greater than 180° otherwise it is called concave.
- A polygon is called regular if all its sides are of equal length and all its angles are equal.
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).
For example, a square is a regular, convex, simple polygon (it is also cyclic).
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. The sum of the inner angles of a simple polygon is equal to (s-2)×180°, where s is the number of its sides.
The area A of a simple 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)
This same formula can also be used to calculate the signed area of complex polygons: follow the sequence of points and count area to the left of your path positive, to the right negative.
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, cyclic polygon, quadrilateral.