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| A '''polygon''' (from the ] ''poly'', for "many", and ''gwnos'', for "angle |
A '''polygon''' (from the ] ''poly'', for "many", and ''gwnos'', for "angle") is a | ||
| closed planar path composed of a finite number of straight lines. | |||
| ⚫ | ''Regular polygons'' have sides that are of equal length and have equal ] between | ||
| succesive pairs of sides. | |||
| ⚫ | ''Regular polygons'' have sides that are of equal length and have equal ] between |
||
| '']'' have at least one internal angle that is greater than 180°, | '']'' have at least one internal angle that is greater than 180°, | ||
| whereas ''convex polygons'' have all internal angles less 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. | ||
| ⚫ | A ''cyclic polygon'' has all of its vertexes lying on the same circle. | ||
| For example, a square is a regular, convex, cyclic polygon. | |||
| ⚫ | A polygon can belong to several classifications simultaneously |
||
| cyclic polygon, for example. | |||
| '''Regular Polygons''' | '''Regular Polygons''' | ||
| <table border=2> | <table border=2> | ||
| <tr> <th>Name</th> <th>Sides</th> <th>Angle*</th> </tr> | <tr> <th>Name</th> <th>Sides</th> <th>Angle*</th> </tr> | ||
| <tr> <td>]</td> <td>3</td> <td>60°</td> </tr> | <tr> <td>]</td> <td>3</td> <td>60°</td> </tr> | ||
| <tr> <td>]</td> <td>4</td> <td>90°</td> </tr> | <tr> <td>]</td> <td>4</td> <td>90°</td> </tr> | ||
| <tr> <td>]</td> <td>5</td> <td>108°</td> </tr> | <tr> <td>]</td> <td>5</td> <td>108°</td> </tr> | ||
| <tr> <td>]</td> <td>6</td> <td>120°</td> </tr> | <tr> <td>]</td> <td>6</td> <td>120°</td> </tr> | ||
| ⚫ | <tr> <td>]</td> <td>7</td> <td>128.57° (approx.)</td> </tr> | ||
| ⚫ | <tr> <td> |
||
| <tr> <td>Octagon</td> <td>8</td> <td>135°</td> </tr> | <tr> <td>Octagon</td> <td>8</td> <td>135°</td> </tr> | ||
| <tr> <td>Nonagon</td> <td>9</td> <td>140°</td> </tr> | <tr> <td>Nonagon</td> <td>9</td> <td>140°</td> </tr> | ||
| <tr> <td>Decagon</td> <td>10</td> <td>144°</td> </tr> | <tr> <td>Decagon</td> <td>10</td> <td>144°</td> </tr> | ||
| <tr> <td>Hectagon</td> <td>100</td> <td>176.4°</td> </tr> | <tr> <td>Hectagon</td> <td>100</td> <td>176.4°</td> </tr> | ||
| <tr> <td>Megagon</td> <td>10<sup>6</sup></td> <td>179.99964°</td> </tr> | <tr> <td>Megagon</td> <td>10<sup>6</sup></td> <td>179.99964°</td> </tr> | ||
| <tr> <td>]</td> <td>10<sup>100</sup></td> <td>180° (approx.)</td> </tr> | <tr> <td>]</td> <td>10<sup>100</sup></td> <td>180° (approx.)</td> </tr> | ||
| </table> | </table> | ||
| * Angle = 180° - 360°<nowiki>/Sides</nowiki> | * Angle = 180° - 360°<nowiki>/Sides</nowiki> | ||
| We will assume ] throughout. | We will assume ] throughout. | ||
| Any polygon, regular or irregular, has as many angles as it has sides, and the sum of its angles | 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. | 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 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>) | ||
| 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.) | :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.) | ||
| ---- | ---- | ||
| See also ], ]. | |||
| ---- | |||
| /Talk | |||
| ] | |||
Revision as of 05:25, 16 February 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. Regular polygons have sides that are of equal length and have equal angles between succesive 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* |
|---|---|---|
| Triangle | 3 | 60° |
| Square | 4 | 90° |
| Pentagon | 5 | 108° |
| Hexagon | 6 | 120° |
| Heptagon | 7 | 128.57° (approx.) |
| Octagon | 8 | 135° |
| Nonagon | 9 | 140° |
| Decagon | 10 | 144° |
| Hectagon | 100 | 176.4° |
| Megagon | 10 | 179.99964° |
| 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.