| Revision as of 15:36, 29 May 2002 view sourceThe Ostrich (talk | contribs)206 edits Proposed taxonomy.← Previous edit | Revision as of 18:00, 29 May 2002 view source AxelBoldt (talk | contribs)Administrators44,505 edits Taxonomy layout, signed areaNext edit → | ||
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| 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 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 |
The taxonomic classification of polygons is illustrated by the following tree: | ||
| <code> | <code> | ||
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| </code> | </code> | ||
| ⚫ | * A polygon is ''simple'' if it is described by a single, non-intersecting boundary; otherwise it is called ''complex''. | ||
| ''Regular polygons'' have sides that are of equal length and have equal ] between successive pairs of sides.<br> | |||
| '' |
* 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. | |||
| ''] polygons'' have at least one internal angle that is greater than 180°.<br> | |||
| ⚫ | '' |
||
| ''Complex polygons'' may have intersecting boundaries.<br> | |||
| 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 (]). | 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 (]). | ||
| 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). | For example, a square is a regular, convex, simple polygon (it is also cyclic). | ||
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| We will assume ] throughout. | We will assume ] throughout. | ||
| Any polygon, regular or irregular, has as many angles as it has sides |
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. | is equal to (''s''-2)×180°, where ''s'' is the number of its sides. | ||
| The ] ''A'' of a polygon can be computed if the cartesian coordinates (''x''<sub>1</sub>, ''y''<sub>1</sub>), (''x''<sub>2</sub>, ''y''<sub>2</sub>), ..., (''x''<sub>''n''</sub>, ''y''<sub>''n''</sub>) of its vertices, listed in order as the area is circulated in counter-clockwise fashion, are known. The formula is | The ] ''A'' of a simple polygon can be computed if the cartesian coordinates (''x''<sub>1</sub>, ''y''<sub>1</sub>), (''x''<sub>2</sub>, ''y''<sub>2</sub>), ..., (''x''<sub>''n''</sub>, ''y''<sub>''n''</sub>) of its vertices, listed in order as the area is circulated in counter-clockwise fashion, are known. The formula is | ||
| :''A'' = 1/2 · (''x''<sub>1</sub>''y''<sub>2</sub> - ''x''<sub>2</sub>''y''<sub>1</sub> + ''x''<sub>2</sub>''y''<sub>3</sub> - ''x''<sub>3</sub>''y''<sub>2</sub> + ... + ''x''<sub>''n''</sub>''y''<sub>1</sub> - ''x''<sub>1</sub>''y''<sub>''n''</sub>) | :''A'' = 1/2 · (''x''<sub>1</sub>''y''<sub>2</sub> - ''x''<sub>2</sub>''y''<sub>1</sub> + ''x''<sub>2</sub>''y''<sub>3</sub> - ''x''<sub>3</sub>''y''<sub>2</sub> + ... + ''x''<sub>''n''</sub>''y''<sub>1</sub> - ''x''<sub>1</sub>''y''<sub>''n''</sub>) | ||
| 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 ruler and compass alone was settled by ] when he was 19: | ||
| 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 18:00, 29 May 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.