| Revision as of 18:10, 27 December 2001 view sourceLee Daniel Crocker (talk | contribs)Extended confirmed users4,417 editsmNo edit summary← Previous edit | Revision as of 19:21, 27 December 2001 view source AxelBoldt (talk | contribs)Administrators44,505 edits "plane" is better than "two-dimensional". +area formula +construction with ruler and compassNext edit → | ||
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| A '''polygon''' is a plane figure that encloses an area | |||
| using straight lines. | using straight lines. | ||
| Regular polygons have sides that are of equal length and have equal ] between each side. | ''Regular polygons'' have sides that are of equal length and have equal ] between each side. | ||
| ] |
'']'' 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 cyclic polygon has all of its vertexes lying on the same circle. | A ''cyclic polygon'' has all of its vertexes lying on the same circle. | ||
| A polygon can belong to several classifications simultaneously; a square is a regular convex | A polygon can belong to several classifications simultaneously; a square is a regular convex | ||
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| * Angle = 180° - 360°<nowiki>/Sides</nowiki> | * Angle = 180° - 360°<nowiki>/Sides</nowiki> | ||
| We will assume ] throughout. | |||
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| 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 | |||
| :''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: | |||
| :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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| /Talk | /Talk | ||
Revision as of 19:21, 27 December 2001
A polygon is a plane figure that encloses an area
using straight lines.
Regular polygons have sides that are of equal length and have equal angles between each side.
Concave polygons have at least one internal angle that is greater than 180°,
whereas convex polygons have all internal angles less than 180°.
A cyclic polygon has all of its vertexes lying on the same circle.
A polygon can belong to several classifications simultaneously; a square is a regular convex
cyclic polygon, for example.
"Poly-" is from the Greek word for "many" and "-gon" is a Greek combining
form meaning "angle".
Regular Polygons
| Name | Sides | Angle* |
|---|---|---|
| Triangle | 3 | 60° |
| Square | 4 | 90° |
| Pentagon | 5 | 108° |
| Hexagon | 6 | 120° |
| Septagon | 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.)
/Talk