POLYGON

POLYGON is an Electronic Warfare Tactics Range located on the border between France and Germany. It is one of only two in Europe, the other being RAF Spadeadam.

Polygon

:''For other use please see Polygon (disambiguation)'' A polygon (literally "many angle", see Wiktionary for the etymology) is a closed curve plane (mathematics) path composed of a finite number of sequential line segments. The straight line segments that make up the polygon are called its ''sides'' or ''edges'' and the points where the sides meet are the polygon's ''vertices''. If a polygon is simple polygon, then its sides (and vertices) constitute the boundary of a polygonal region, and the term ''polygon'' sometimes also describes the ''interior'' of the polygonal region (the open area that this path encloses) or the union of both the region and its boundary. == Names and types ==

Polygons are named according to the number of sides, combining a Greek language-derived numerical prefix with the suffix ''-gon'', e.g. ''pentagon'', ''dodecagon''. The triangle (geometry) and quadrilateral are exceptions. For larger numbers, mathematicians write the numeral itself, e.g. ''17-gon''. A variable can even be used, usually ''n-gon''. This is useful if the number of sides is used in a formula. {| |- |+ Polygon names |- ! Name !! Sides |- | triangle (geometry) (or trigon) || 3 |- | quadrilateral (or tetragon) || 4 |- | pentagon || 5 |- | hexagon || 6 |- | heptagon (avoid "septagon") || 7 |- | octagon || 8 |- | enneagon (or "nonagon") || 9 |- | decagon || 10 |- | hendecagon (avoid "undecagon") || 11 |- | dodecagon (avoid "duodecagon") || 12 |- | triskaidecagon || 13 |- | pentadecagon || 15 |- | heptadecagon || 17 |- | enneadecagon || 19 |- | icosagon || 20 |- | triacontagon || 30 |- | pentacontagon || 50 |- | hectagon (avoid "centagon") || 100 |- | chiliagon || 1000 |- | myriagon || 10,000 |} === Naming polygons === To construct the name of a polygon with more than 20 and less than 100 sides, combine the prefixes as follows {| style="vertical-align:center;" |- style="text-align:center;" ! colspan="2" rowspan="2" | Tens ! ''and'' ! colspan="2" | Ones ! final prefix |- ! rowspan="9" | -kai- | 1 | hena- ! rowspan=9 | -gon |- | 20 || icosi- || 2 || -di- |- | 30 || triaconta- || 3 || -tri- |- | 40 || tetraconta- || 4 || -tetra- |- | 50 || pentaconta- || 5 || -penta- |- | 60 || hexaconta- || 6 || -hexa- |- | 70 || heptaconta- || 7 || -hepta- |- | 80 || octaconta- || 8 || -octa- |- | 90 || enneaconta- || 9 || -ennea- |- |} That is, a 42-sided figure would be named as follows: {| |- ! Tens ! ''and'' ! Ones ! final prefix ! full polygon name |- | tetraconta- | -kai- | -di- | -gon | tetracontakaidigon |- |} and a 50-sided figure {| |- ! Tens ! ''and'' ! Ones ! final prefix ! full polygon name |- | pentaconta- | colspan="2"| | -gon | pentacontagon |- |} But beyond nonagons and decagons, professional mathematicians prefer the aforementioned numeral notation (for example, MathWorld has articles on 17-gons and 257-gons). == Taxonomic classification == The taxonomic classification of polygons is illustrated by the following tree:


                                       Polygon
                                      /       \
                                  Simple     Complex
                                 /     \
                            Convex     Concave
                             /
                        Cyclic 
                        /    
                   Regular

* A polygon is called ''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 convex polygon is called concyclic or cyclic polygon if all the vertices lie on a single circle. * A cyclic polygon is called ''regular'' if all its sides are of equal length and all its angles are equal. == Properties == We will assume Euclidean geometry throughout. === Angles === 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)Pi radians (or (''n''−2)180�), and the inner angle of a regular ''n''-gon is (''n''−2)π/''n'' radians (or (''n''−2)180�/''n''). This can be seen in two different ways: * Moving around a simple ''n''-gon (like a car on a road), the amount one "turns" at a vertex is 180� minus the inner angle. "Driving around" the polygon, one makes one full turn, so the sum of these turns must be 360�, from which the formula follows easily. The reasoning also applies if some inner angles are more than 180�: going clockwise around, it means that one sometime turns left instead of right, which is counted as a negative amount one turns. * Any simple ''n''-gon can be considered to be made up of (''n''−2) triangles, each of which has an angle sum of π radians or 180�. === Area === [[Image:Apothem_of_hexagon.png|thumb|right|Apothem of an hexagon]] Several formulae give the area of a regular polygon: :

A=\frac{nt^2}{4\tan(180^\circ/n)}

: half the perimeter multiplied by the length of the apothem (the line drawn from the centre of the polygon perpendicular to a side) The area (geometry) ''A'' of a simple polygon can be computed if the cartesian coordinate system (''x''₁, ''y''₁), (''x''₂, ''y''₂), ..., (''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'' = � � (''x''₁''y''₂ − ''x''₂''y''₁ + ''x''₂''y''₃ − ''x''₃''y''₂ + ... + ''x''_''n''''y''₁ − ''x''₁''y''_''n'') : = � � (''x''₁(''y''₂ − ''y''_''n'') + ''x''₂(''y''₃ − ''y''₁) + ''x''₃(''y''₄ − ''y''₂) + ... + ''x''_''n''(''y''₁ − ''y''_''n''−1)) The formula was described by Meister in 1769 and by Carl Friedrich Gauss in 1795. It can be verified by dividing the polygon into triangles, but it can also be seen as a special case of Green's theorem. If the polygon can be drawn on an equally-spaced grid such that all its vertices are grid points, Pick's theorem gives a simple formula for the polygon's area based on the numbers of interior and boundary grid points. If any two simple polygons of equal area are given, then the first can be cut into polygonal pieces which can be reassembled to form the second polygon. This is the Bolyai-Gerwien theorem. === Construction === All regular polygons are concyclic, as are all triangles and rectangles (see circumcircle). A regular ''n''-sided polygon can be constructed with Ruler-and-compass construction if and only if the odd number prime number factors of ''n'' are distinct Fermat primes. See constructible polygon. == Point in polygon test == In computer graphics and computational geometry, it is often necessary to determine whether a given point ''P'' = (''x''₀,''y''₀) lies inside a simple polygon given by a sequence of line segments. It is known as Point in polygon test. == See also == * cyclic polygon * geometric shape * polyform * polyhedron * polytope * simple polygon * synthetic geometry * tiling * tiling puzzle Polygons vi:Đa gi�c

Polygon

AxelBoldt removed the statement: Strictly speaking, every polyhedron is also a polygon as is every polytope, since they all have angles. with the simple claim: Polyhedra are not polygons. Since it is obvious that polyhedron have multiple angles, and hence are polygonal, I'd like to give him a chance to explain why he removed true information. ---- Simple: because it's not true at all. "Polygon" is almost universally defined as a 2-dimensional figure. I don't know of any mathematics text or course that treats it as a superclass that includes 3-d polyhedra. Terms here should be used as they are commonly used in academia. --User:Lee Daniel Crocker ---- Lee, perhaps you would like to suggest what term should be used for the class of objects that have multiple angles, regardless of the dimentionality? Then we could put in a reference to that class of objects in this article. I had never heard the term polytope before I got involved with Wikipedia. Does the concept of the angle between two planes make sense? I, admittedly, have found very little accessible material on these topics. -- BenBaker Maybe a phrase such as: Even though strictly speaking, every polyhedron has multiple angles, as does every polytope, they are not considered as polygons as the angles between their faces are not two dimensional. They can be classified as 'technical-term', however. :"Polytope" ''is'' the general term, although it is typically only used to refer to 4-d and higher figures (because the 2- and 3-d figures already have names). It is, nonetheless, proper to refer to polygons and polyhedra as subclasses of polytopes. --LDC ---- Mathematical terms are not defined etymologically. "Polygon" may mean "many angles" in Greek, but that doesn't mean that anything with many angles is called a polygon in mathematics (and yes, you can have :angle between planes). Polygons are two-dimensional figures that enclose an area with straight lines. We could have links to polytopes and polyhedra I suppose. --AxelBoldt ---- I am not aware of ''any'' word in ''any'' context that is defined by its etymology. Words mean whatever they are defined to mean, regardless of where they happen to come from. Adding an explicit statement to that effect in this article would be silly, because that's just a case of understanding the nature of the English language and has nothing to do with polygons. This article is about polygons, which are flat. Now, if you want to add some statement to the effect that polygons are the two-dimensional instance of the more general class of polytopes, that's entirely appropriate. --LDC ------ "Words mean whatever they are defined to mean, regardless of where they happen to come from." -- Indeed. That's the Humpty Dumpty argument! (Through the Looking-Glass) I'd like to mention the term "n-gon" on this page, since that's the link I followed to get here. Also, the table is somewhat inconsistent: a "Triangle" may be regular or not. A "Square" is regular by definition. The other terms ''usually'' are taken to mean the regular form -- in my experience it's more common to see a phrase like "an irregular pentagon" than "a regular pentagon". :Fixed. --user:Damian Yerrick On the dimension issue, it might be fair to mention that a polygon is a 2D polytope, but it's not terribly interesting. The question of a "broken" polygon in higher dimensions -- ie a set of non-planar points joined by a closed, simple path -- is perhaps interesting, but completely breaks the definition of a polytope as a convex hull of point, and there's no longer any notion of area or volume. I suppose then it's merely a path. -- Tarquin ------ Hi. I added a proposed taxonomy, but it does have problems. There is the problem that under the definitions that I left, a complex polygon may be considered convex. Is this indeed the case? If so, the two versions of convex are surely nevetheless considered distinct, so Simple convex and complex convex are distinct classes, both denoted 'convex'? Or am I just being too hopeful in proposing a tree-based taxonomy? ---- I would prefer a more simple and complete description of the sum of inner angles in an n-gon. Allthough the one used looks simple, it is based on foreknowledge of the sum of angles in a triangle. I sugest something like the following description(better phrased probably). (For lack of a better word or expression(i can't seem to find it), i will use the term ''outer angle'' for the outside coresponding angle which is 180 - inner angle.)
The sum of the outer angles of an n-gon is a full circle.
The average size of an outer angle is therefore 360/n.
The average size of an inner angle is therefore 180 - (360/n).
The sum of inner angles is therefore n * (180 - (360/n)) <=> 180n - 360 <=> 180 * (n-2).
(exscuse the language, english is not my native tongue). User:Jan Pedersen 09:19 21 Jul 2003 (UTC) :I added something along these lines. - User:Patrick 09:48 21 Jul 2003 (UTC) ---- :''its vertices, listed in order as the area is circulated in counter-clockwise fashion,'' Is "circulated" the right verb here? User:AxelBoldt 21:48, 26 Sep 2003 (UTC) Mathematicians are notoriously incompetent historians. Gauss NEVER gave a proof of the necessity of the constructibility of the regular ''n''-gon. WANTZEL proved this in 1837. If you read otherwise, it's because mathematicians are sloppy historians. user:revolver == Names of Polygons?? == *A hectagon has 100 sides. *A chiliagon has 1000 sides. *A myriagon has 10,000 sides. *A "megagon" has 1,000,000 sides. How about 100,000 sides?? User:66.245.71.11 21:53, 1 May 2004 (UTC) Is it necessary to have "googolgon"? The word is an irregular formation and there is no evidence of its ever having been used. == Enneagon/nonagon == The table of names of polygons specifies for a nine-sided polygon the name "enneagon" and then admonishes "(avoid 'nonagon')". The discussion further down includes the sentence, "But beyond nonagons and decagons, professional mathematicians prefer the aforementioned numeral notation." Personally, I think "nonagon" is in sufficiently widespread use to be acceptable even if inconsistent -- much the same way that "quadrilateral" is more widely accepted than "tetragon". So I'd prefer to see the table changed than the sentence in the discussion. How do other people feel? -Heath === Response === To whoever wrote the above discussion, here are some comments. First, when we sign Wikipedia articles, you don't write a word like "Heath"; you just write ~~~~ . I've studied the history of the talk page and found out that you are not a registered Wikipedian. Second, it already says just above the table that "the triangle and quadrilateral are exceptions", as well as a comment saying "(or tetragon)" meaning that quadrilateral is a more well-known word than tetragon, but that both words are equally proper. Third, the info on ennea vs. nona as the prefix for 9 is already mentioned at the bottom of the Greek numerical prefixes article, saying that "In practice, people often use Latin nona- for 9 instead of Greek ennea-." User:Georgia guy 21:11, 2 Feb 2005 (UTC) === Response === Thanks for the note on the Wiki etiquette -- I'm not (yet) a registered Wikipedian; still learning my way but trying to be constructive in the process. ~~~~ is a tip I hadn't yet seen. The point that I was trying to make originally is that the table specifically says to "avoid" the word ''nonagon'', while further down the same page the article actually uses that word. The Greek numerical prefixes article does comment further on this, but someone reading ''this'' article for the first time is not likely to have seen that other article (I sure hadn't!), and is likely to find the usage in this article contradictory to its own recommendations. User:128.173.105.144 03:39, 8 Feb 2005 (UTC) (Heath) == Names of Polygons == Currently, several Wikipedia articles on polygons between 20 and 100 sides are on Vfd and I want to see if anyone plans on creating a Names of Polygons article that is similar to the Names of large numbers article. Anyone in favor of doing so?? User:Georgia guy 22:56, 10 Feb 2005 (UTC) == Vfd consensus == What is the Vfd consensus of the polygon articles put on Vfd about a week ago?? Has it been reached yet?? User:Georgia guy 20:27, 18 Feb 2005 (UTC) == "Moving around a simple n-gon" == This is only a very small problem but I didn't think I should make the edit myself just in case. When I first read this I thought it was talking about moving the whole polygon around. I quickly realised what it really meant, but thought it should be slightly clarified. Moving around on the edges of a simple n-gon There could also be a link on edges if anyone changes it.

See other meanings of words starting from letter:

P

PA | PB | PC | PD | PE | PF | PG | PH | PI | PJ | PK | PL | PM | PN | PO | PR | PS | PT | PU | PW | PX | PY | PZ |

Words begining with Polygon:

POLYGON
Polygon
Polygon
Polygonaceae
Polygonaceous
Polygonal
Polygonales
Polygonal_number
Polygonal_number
Polygonal_tiling
Polygonatum
Polygonia_c-album
Polygons
Polygons
Polygons
Polygonum
Polygonum_convolvulus
Polygonum_cuspidatum
Polygonum_odoratum
Polygonum_perfoliatum
Polygonum_persicaria
Polygonum_viviparum
Polygon_(computer_graphics)
Polygon_(disambiguation)
Polygon_(disambiguation)
Polygon_area
Polygon_count
Polygon_mesh
Polygon_modeling
Polygon_Records
Polygon_triangulation
Polygon_Window