Polygon: Difference between revisions

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Revision as of 14:13, 29 May 2002 view sourceDamian Yerrick (talk \| contribs)Autopatrolled, Extended confirmed users, Pending changes reviewers15,087 edits Clarified regular polys section← Previous edit		Revision as of 15:36, 29 May 2002 view source The Ostrich (talk \| contribs)206 edits Proposed taxonomy.Next edit →
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	closed planar path composed of a finite number of straight lines.		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 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 somewhat complex, but the following tree may shed some light on the matter:

			<code>
			Polygon
			/ \
			Simple Complex
			/ \
			Convex Concave
			/
			Regular
			</code>

	''Regular polygons'' have sides that are of equal length and have equal ] between successive pairs of sides.<br>		''Regular polygons'' have sides that are of equal length and have equal ] between successive pairs of sides.<br>
	'']~~'' have at least one internal angle that is greater than 180°, whereas ''convex~~ polygons'' have ~~all~~ internal angles ~~less~~ than 180°.<br>		''] polygons'' have no internal angles greater than 180&deg.<br>
			''] polygons'' have at least one internal angle that is greater than 180&deg.<br>
	A ''concyclic'' or ''cyclic polygon'' has all of its vertexes lying on the same circle.<br>
			''Simple polygons'' are described by a single, non-intersecting boundary.<br>
	A polygon can belong to several classifications simultaneously.
			''Complex polygons'' may have intersecting boundaries.<br>
⚫	For example, a square is a regular, convex, ~~cyclic~~ polygon.

			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).

	'''Regular Polygons'''		'''Regular Polygons'''

Revision as of 15:36, 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 somewhat complex, but the following tree may shed some light on the matter:

                                      Polygon
                                     /       \
                                 Simple     Complex
                                /     \
                           Convex     Concave
                            /
                       Regular

Regular polygons have sides that are of equal length and have equal angles between successive pairs of sides.
Convex polygons have no internal angles greater than 180&deg.
Concave polygons have at least one internal angle that is greater than 180&deg.
Simple polygons are described by a single, non-intersecting boundary.
Complex polygons may have intersecting boundaries.

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).

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).

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, 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), ..., (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)

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.)