Polygon: Difference between revisions

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	We will assume ] throughout.		We will assume ] throughout.

	All regular polygons are ~~cyclic~~, as are all triangles (see ]).		All regular polygons are concyclic, as are all triangles (see ]).

	Any polygon, regular or irregular, complex or simple, has as many angles as it has sides. The sum of the inner ~~angles~~ of a simple ''n''-gon is (n-2)], and the inner angle of a regular ''n''-gon is π - 2π / ''n''.		Any polygon, regular or irregular, complex or simple, has as many angles as it has sides. The sum of the inner ]s of a simple ''n''-gon is (n-2)] ]s, and the inner angle of a regular ''n''-gon is π - 2π / ''n''.

	<table border=2>		<table border=2>
	<tr><th>~~Name</th>~~ ~~<th>Sides~~</th>~~<th>Angle*</th>~~ </tr>		<tr><th colspan=3>Regular Polygons</th></tr>
			<tr><th>Name</th> <th>Sides</th><th>Angle</th> </tr>
	<tr><td>equilateral ]</td><td>3</td> <td>60°</td> </tr>		<tr><td>equilateral ]</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>
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	</table>		</table>

	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		The ] ''A'' of a simple polygon can be computed if the ] (''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.		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 ] 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 ruler and compass if and only if the odd ] factors of ''n'' are distinct prime numbers of the form		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

Revision as of 16:46, 11 February 2003

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.

Names and types

Polygons are named according to the number of sides, combining a Greek root with the suffix -gon, eg pentagon, dodecagon. The triangle and quadrilateral are exceptions. For larger numbers, mathematicians write the numeral itself, eg 17-gon. A variable can even be used, usually n-gon. This is useful if the number of sides is used in a formula.

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.

For example, a square is a regular, cyclic 4-gon.

Properties

We will assume Euclidean geometry throughout.

All regular polygons are concyclic, as are all triangles (see circumcircle).

Any polygon, regular or irregular, complex or simple, has as many angles as it has sides. The sum of the inner angles of a simple n-gon is (n-2)π radians, and the inner angle of a regular n-gon is π - 2π / n.

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.)

The area A of a simple polygon can be computed if the cartesian coordinates (x₁, y₁), (x₂, y₂), ..., (x_n, y_n) 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 + ... + x_ny₁ - x₁y_n)

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 Carl Friedrich 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^{n}}+1

The only known primes of this type are 3, 5, 17, 257, 65537: the Fermat primes.

See also: geometric shape, polyhedron, polytope, cyclic polygon.