Editing Polygon (section)

==Properties and formulas==
[[File:Winkelsumme-polygon.svg|thumb|upright=1.0|Partitioning an ''n''-gon into {{nowrap|''n'' − 2}} triangles]]
[[Euclidean geometry]] is assumed throughout.

===Angles===
Any polygon has as many corners as it has sides. Each corner has several angles. The two most important ones are:
* '''[[Interior angle]]''' – The sum of the interior angles of a simple ''n''-gon is {{nowrap|(''n'' − 2) × [[Pi|π]]}} [[radian]]s or {{nowrap|(''n'' − 2) × 180}} [[degree (angle)|degrees]]. This is because any simple ''n''-gon ( having ''n'' sides ) can be considered to be made up of {{nowrap|(''n'' − 2)}} triangles, each of which has an angle sum of π radians or 180 degrees. The measure of any interior angle of a convex regular ''n''-gon is <math>\left(1-\tfrac{2}{n}\right)\pi</math> radians or <math>180-\tfrac{360}{n}</math> degrees. The interior angles of regular [[star polygon]]s were first studied by Poinsot, in the same paper in which he describes the four [[Kepler–Poinsot polyhedron|regular star polyhedra]]: for a regular <math>\tfrac{p}{q}</math>-gon (a ''p''-gon with central density ''q''), each interior angle is <math>\tfrac{\pi(p-2q)}{p}</math> radians or <math>\tfrac{180(p-2q)}{p}</math> degrees.<ref>{{cite book |last=Kappraff |first=Jay |title=Beyond measure: a guided tour through nature, myth, and number |publisher=World Scientific |year=2002 |page=258 |isbn= 978-981-02-4702-7 |url=https://books.google.com/books?id=vAfBrK678_kC&q=star+polygon&pg=PA256}}</ref>
* '''[[Exterior angle]]''' – The exterior angle is the [[supplementary angle]] to the interior angle. Tracing around a convex ''n''-gon, the angle "turned" at a corner is the exterior or external angle. Tracing all the way around the polygon makes one full [[Turn (geometry)|turn]], so the sum of the exterior angles must be 360°. This argument can be generalized to concave simple polygons, if external angles that turn in the opposite direction are subtracted from the total turned. Tracing around an ''n''-gon in general, the sum of the exterior angles (the total amount one rotates at the vertices) can be any integer multiple ''d'' of 360°, e.g. 720° for a [[pentagram]] and 0° for an angular "eight" or [[antiparallelogram]], where ''d'' is the [[Density (polytope)#Polygons|density]] or [[turning number]] of the polygon.

===Area===
[[File:Polygon vertex labels.svg|thumb|320px|right|Coordinates of a non-convex pentagon]]

In this section, the vertices of the polygon under consideration are taken to be <math>(x_0, y_0), (x_1, y_1), \ldots, (x_{n - 1}, y_{n - 1})</math> in order. For convenience in some formulas, the notation {{math|1=(''x<sub>n</sub>'', ''y<sub>n</sub>'') = (''x''<sub>0</sub>, ''y''<sub>0</sub>)}} will also be used.

====Simple polygons====
{{further|Shoelace formula}}

If the polygon is non-self-intersecting (that is, [[simple polygon|simple]]), the signed [[area (geometry)|area]] is
:<math>A = \frac{1}{2} \sum_{i = 0}^{n - 1}( x_i y_{i + 1} - x_{i + 1} y_i) \quad \text {where } x_{n}=x_{0} \text{ and } y_n=y_{0}, </math>
or, using [[determinant]]s
:<math>16 A^{2} = \sum_{i=0}^{n-1} \sum_{j=0}^{n-1} \begin{vmatrix} Q_{i,j} & Q_{i,j+1} \\
Q_{i+1,j} & Q_{i+1,j+1} \end{vmatrix} , </math>
where <math> Q_{i,j} </math> is the squared distance between <math>(x_i, y_i)</math> and <math>(x_j, y_j).</math><ref>B.Sz. Nagy, L. Rédey: Eine Verallgemeinerung der Inhaltsformel von Heron. Publ.
Math. Debrecen 1, 42–50 (1949)</ref><ref>{{cite web
 |url      = http://www.seas.upenn.edu/~sys502/extra_materials/Polygon%20Area%20and%20Centroid.pdf
 |title      = Calculating The Area And Centroid Of A Polygon
 |last      = Bourke
 |first      = Paul
 |date      = July 1988
 |access-date      = 6 Feb 2013
 |archive-date      = 16 September 2012
 |archive-url      = https://web.archive.org/web/20120916104133/http://www.seas.upenn.edu/~sys502/extra_materials/Polygon%20Area%20and%20Centroid.pdf
 |url-status      = dead
 }}</ref>

The signed area depends on the ordering of the vertices and of the [[orientation (vector space)|orientation]] of the plane. Commonly, the positive orientation is defined by the (counterclockwise) rotation that maps the positive {{mvar|x}}-axis to the positive {{mvar|y}}-axis. If the vertices are ordered counterclockwise (that is, according to positive orientation), the signed area is positive; otherwise, it is negative. In either case, the area formula is correct in [[absolute value]]. This is commonly called the ''[[shoelace formula]]'' or ''surveyor's formula''.<ref>{{cite journal |author=Bart Braden |title=The Surveyor's Area Formula |journal=The College Mathematics Journal |volume=17 |issue=4 |year=1986 |pages=326–337 |url=http://www.maa.org/pubs/Calc_articles/ma063.pdf|archive-url=https://web.archive.org/web/20121107190918/http://www.maa.org/pubs/Calc_articles/ma063.pdf|archive-date=2012-11-07 |doi=10.2307/2686282|jstor=2686282 }}</ref>

The area ''A'' of a simple polygon can also be computed if the lengths of the sides, ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a<sub>n</sub>'' and the [[exterior angle]]s, ''θ''<sub>1</sub>, ''θ''<sub>2</sub>, ..., ''θ<sub>n</sub>'' are known, from:
:<math>\begin{align}A = \frac12 ( a_1[a_2 \sin(\theta_1) + a_3 \sin(\theta_1 + \theta_2) + \cdots + a_{n-1} \sin(\theta_1 + \theta_2 + \cdots + \theta_{n-2})] \\
{} + a_2[a_3 \sin(\theta_2) + a_4 \sin(\theta_2 + \theta_3) + \cdots + a_{n-1} \sin(\theta_2 + \cdots + \theta_{n-2})] \\
{} + \cdots + a_{n-2}[a_{n-1} \sin(\theta_{n-2})] ). \end{align}</math>
The formula was described by Lopshits in 1963.<ref name="lopshits">{{cite book |title=Computation of areas of oriented figures |author=A.M. Lopshits |publisher=D C Heath and Company: Boston, MA |others=translators: J Massalski and C Mills Jr. |year=1963}}</ref>

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: the former number plus one-half the latter number, minus 1.

In every polygon with perimeter ''p'' and area ''A '', the [[isoperimetric inequality]] <math>p^2 > 4\pi A</math> holds.<ref>{{cite web| url = http://forumgeom.fau.edu/FG2002volume2/FG200215.pdf| title = Dergiades, Nikolaos, "An elementary proof of the isoperimetric inequality", ''Forum Mathematicorum'' 2, 2002, 129–130.}}</ref>

For any two simple polygons of equal area, the [[Bolyai–Gerwien theorem]] asserts that the first can be cut into polygonal pieces which can be reassembled to form the second polygon.

The lengths of the sides of a polygon do not in general determine its area.<ref>Robbins, "Polygons inscribed in a circle", ''American Mathematical Monthly'' 102, June–July 1995.</ref> However, if the polygon is simple and cyclic then the sides ''do'' determine the area.<ref>{{cite journal|last=Pak|first=Igor|author-link=Igor Pak|doi=10.1016/j.aam.2004.08.006|issue=4|journal=[[Advances in Applied Mathematics]]|mr=2128993|pages=690–696|title=The area of cyclic polygons: recent progress on Robbins' conjectures|volume=34|year=2005|arxiv=math/0408104|s2cid=6756387}}</ref> Of all ''n''-gons with given side lengths, the one with the largest area is cyclic. Of all ''n''-gons with a given perimeter, the one with the largest area is regular (and therefore cyclic).<ref>Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in ''Mathematical Plums'' (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.</ref>

====Regular polygons====
Many specialized formulas apply to the areas of [[regular polygon]]s.

The area of a regular polygon is given in terms of the radius ''r'' of its [[inscribed circle]] and its perimeter ''p'' by 
:<math>A = \tfrac{1}{2} \cdot p \cdot r.</math>
This radius is also termed its [[apothem]] and is often represented as ''a''.

The area of a regular ''n''-gon can be expressed in terms of the radius ''R'' of its [[circumscribed circle]] (the unique circle passing through all vertices of the regular ''n''-gon) as follows:<ref>[https://www.mathopenref.com/polygonregularareaderive.html Area of a regular polygon – derivation] from Math Open Reference.</ref><ref>A regular polygon with an infinite number of sides is a circle: <math>\lim_{n \to +\infty} R^2 \cdot \frac{n}{2} \cdot \sin \frac{2\pi}{n} = \pi \cdot R^2</math>.</ref>
:<math>A = R^2 \cdot \frac{n}{2} \cdot \sin \frac{2\pi}{n} = R^2 \cdot n \cdot \sin \frac{\pi}{n} \cdot \cos \frac{\pi}{n}</math>

====Self-intersecting====
The area of a [[Complex polygon|self-intersecting polygon]] can be defined in two different ways, giving different answers:
* Using the formulas for simple polygons, we allow that particular regions within the polygon may have their area multiplied by a factor which we call the ''density'' of the region. For example, the central convex pentagon in the center of a pentagram has density 2. The two triangular regions of a cross-quadrilateral (like a figure 8) have opposite-signed densities, and adding their areas together can give a total area of zero for the whole figure.<ref>{{cite journal|url=http://dynamicmathematicslearning.com/crossed-quad-area.pdf|title=Slaying a geometrical 'Monster': finding the area of a crossed Quadrilateral|last=De Villiers|first=Michael|journal=Learning and Teaching Mathematics|volume=2015|issue=18|date=January 2015|pages=23–28}}</ref>
* Considering the enclosed regions as point sets, we can find the area of the enclosed point set. This corresponds to the area of the plane covered by the polygon or to the area of one or more simple polygons having the same outline as the self-intersecting one. In the case of the cross-quadrilateral, it is treated as two simple triangles.{{citation needed|date=February 2019}}

===Centroid===
Using the same convention for vertex coordinates as in the previous section, the coordinates of the centroid of a solid simple polygon are 
:<math>C_x = \frac{1}{6 A} \sum_{i = 0}^{n - 1} (x_i + x_{i + 1}) (x_i y_{i + 1} - x_{i + 1} y_i), </math>
:<math>C_y = \frac{1}{6 A} \sum_{i = 0}^{n - 1} (y_i + y_{i + 1}) (x_i y_{i + 1} - x_{i + 1} y_i).</math>
In these formulas, the signed value of area <math>A</math> must be used.

For [[triangle]]s ({{math|1=''n'' = 3}}), the centroids of the vertices and of the solid shape are the same, but, in general, this is not true for {{math|''n'' > 3}}. The [[centroid]] of the vertex set of a polygon with {{mvar|n}} vertices has the coordinates
:<math>c_x=\frac 1n \sum_{i = 0}^{n - 1}x_i,</math>
:<math>c_y=\frac 1n \sum_{i = 0}^{n - 1}y_i.</math>