Editing Polygon (section)

===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}}