Editing Area (section)

==Area formulas==

===Polygon formulas===
{{Main article|Polygon#Area}}

For a non-self-intersecting ([[simple polygon|simple]]) polygon, the [[Cartesian coordinate system|Cartesian coordinates]] <math>(x_i,  y_i)</math> (''i''=0, 1, ..., ''n''-1) of whose ''n'' [[vertex (geometry)|vertices]] are known, the area is given by the [[shoelace formula|surveyor's formula]]:<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
 |url-status     = live
 |archive-url  = https://web.archive.org/web/20120916104133/http://www.seas.upenn.edu/~sys502/extra_materials/Polygon%20Area%20and%20Centroid.pdf
 |archive-date = 2012-09-16
}}
</ref>

:<math>A = \frac{1}{2} \Biggl\vert \sum_{i = 0}^{n - 1}( x_i y_{i + 1} - x_{i + 1} y_i) \Biggr\vert</math>

where when ''i''=''n''-1, then ''i''+1 is expressed as [[modular arithmetic|modulus]] ''n'' and so refers to 0.

====Rectangles====
[[File:RectangleLengthWidth.svg|thumb|right|upright|alt=A rectangle with length and width labelled|The area of this rectangle is&nbsp;{{mvar|lw}}.]]
The most basic area formula is the formula for the area of a [[rectangle]]. Given a rectangle with length {{mvar|l}} and width {{mvar|w}}, the formula for the area is:<ref name=AF>{{cite web |url=http://www.math.com/tables/geometry/areas.htm |title=Area Formulas |publisher=Math.com |access-date=2 July 2012 |url-status=live |archive-url=https://web.archive.org/web/20120702135710/http://www.math.com/tables/geometry/areas.htm |archive-date=2 July 2012}}</ref>

:{{bigmath|''A'' {{=}} ''lw''}} &nbsp;(rectangle).
That is, the area of the rectangle is the length multiplied by the width.  As a special case, as {{math|''l'' {{=}} ''w''}} in the case of a square, the area of a square with side length {{mvar|s}} is given by the formula:<ref name=MathWorld/><ref name=AF/>
:{{bigmath|''A'' {{=}} ''s''<sup>2</sup>}} &nbsp;(square).

The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a [[definition]] or [[axiom]].  On the other hand, if [[geometry]] is developed before [[arithmetic]], this formula can be used to define [[multiplication]] of [[real number]]s.

====Dissection, parallelograms, and triangles====
{{Main article|Triangle area|Parallelogram#Area formula}}
[[File:ParallelogramArea.svg|thumb|right|upright|A parallelogram can be cut up and re-arranged to form a rectangle.]]

Most other simple formulas for area follow from the method of [[dissection (geometry)|dissection]].
This involves cutting a shape into pieces, whose areas must [[addition|sum]] to the area of the original shape.
For an example, any [[parallelogram]] can be subdivided into a [[trapezoid]] and a [[right triangle]], as shown in figure to the left.  If the triangle is moved to the other side of the trapezoid, then the resulting figure is a rectangle.  It follows that the area of the parallelogram is the same as the area of the rectangle:<ref name=AF/>
:{{bigmath|''A'' {{=}} ''bh''}} &nbsp;(parallelogram).
[[File:TriangleArea.svg|thumb|right|upright|A parallelogram split into two equal triangles]]
However, the same parallelogram can also be cut along a [[diagonal]] into two [[congruence (geometry)|congruent]] triangles, as shown in the figure to the right.  It follows that the area of each [[triangle]] is half the area of the parallelogram:<ref name=AF/>
:<math>A = \frac{1}{2}bh</math> &nbsp;(triangle).
Similar arguments can be used to find area formulas for the [[trapezoid]]<ref>{{citation|title=Problem Solving Through Recreational Mathematics|title-link=Problem Solving Through Recreational Mathematics|first1=Bonnie|last1=Averbach|author1-link=Bonnie Averbach|first2=Orin|last2=Chein|publisher=Dover|year=2012|isbn=978-0-486-13174-0|page=[https://books.google.com/books?id=Dz_CAgAAQBAJ&pg=PA306 306]}}</ref> as well as more complicated [[polygon]]s.<ref>{{citation|title=Calculus for Scientists and Engineers: An Analytical Approach|first=K. D.|last=Joshi|publisher=CRC Press|year=2002|isbn=978-0-8493-1319-6|page=43|url=https://books.google.com/books?id=5SDcLHkelq4C&pg=PA43|url-status=live|archive-url=https://web.archive.org/web/20160505011253/https://books.google.com/books?id=5SDcLHkelq4C&pg=PA43|archive-date=2016-05-05}}</ref>

===Area of curved shapes===

====Circles====
[[File:CircleArea.svg|thumb|right|alt=A circle divided into many sectors can be re-arranged roughly to form a parallelogram|A circle can be divided into [[Circular sector|sectors]] which rearrange to form an approximate [[parallelogram]].]]
{{main article|Area of a circle}}
The formula for the area of a [[circle]] (more properly called the area enclosed by a circle or the area of a [[disk (mathematics)|disk]]) is based on a similar method.  Given a circle of radius {{math|''r''}}, it is possible to partition the circle into [[Circular sector|sectors]], as shown in the figure to the right.  Each sector is approximately triangular in shape, and the sectors can be rearranged to form an approximate parallelogram.  The height of this parallelogram is {{math|''r''}}, and the width is half the [[circumference]] of the circle, or {{math|π''r''}}.  Thus, the total area of the circle is {{math|π''r''<sup>2</sup>}}:<ref name=AF/>
:{{bigmath|''A'' {{=}} π''r''<sup>2</sup>}} &nbsp;(circle).
Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors.  The [[limit (mathematics)|limit]] of the areas of the approximate parallelograms is exactly {{math|π''r''<sup>2</sup>}}, which is the area of the circle.<ref name=Surveyor/>

This argument is actually a simple application of the ideas of [[calculus]].  In ancient times, the [[method of exhaustion]] was used in a similar way to find the area of the circle, and this method is now recognized as a precursor to [[integral calculus]].  Using modern methods, the area of a circle can be computed using a [[definite integral]]:
:<math>A \;=\;2\int_{-r}^r \sqrt{r^2 - x^2}\,dx \;=\; \pi r^2.</math>

====Ellipses====
{{main article|Ellipse#Area}}
The formula for the area enclosed by an [[ellipse]] is related to the formula of a circle; for an ellipse with [[semi-major axis|semi-major]] and [[semi-minor axis|semi-minor]] axes {{math|''x''}} and {{math|''y''}} the formula is:<ref name=AF/>
:<math>A = \pi xy .</math>

===Non-planar surface area===
{{main article|Surface area}}
[[File:Archimedes sphere and cylinder.svg|right|thumb|alt=A blue sphere inside a cylinder of the same height and radius|[[Archimedes]] showed that the surface area of a [[sphere]] is exactly four times the area of a flat [[disk (mathematics)|disk]] of the same radius, and the volume enclosed by the sphere is exactly 2/3 of the volume of a [[cylinder (geometry)|cylinder]] of the same height and radius.]]

Most basic formulas for [[surface area]] can be obtained by cutting surfaces and flattening them out (see: [[developable surface]]s).  For example, if the side surface of a [[cylinder (geometry)|cylinder]] (or any [[prism (geometry)|prism]]) is cut lengthwise, the surface can be flattened out into a rectangle.  Similarly, if a cut is made along the side of a [[cone (geometry)|cone]], the side surface can be flattened out into a sector of a circle, and the resulting area computed.

The formula for the surface area of a [[sphere]] is more difficult to derive: because a sphere has nonzero [[Gaussian curvature]], it cannot be flattened out.  The formula for the surface area of a sphere was first obtained by [[Archimedes]] in his work ''[[On the Sphere and Cylinder]]''.  The formula is:<ref name=MathWorldSurfaceArea/>
:{{math|''A'' {{=}} 4''πr''<sup>2</sup>}} &nbsp;(sphere), 
where {{math|''r''}} is the radius of the sphere.  As with the formula for the area of a circle, any derivation of this formula inherently uses methods similar to [[calculus]].

===General formulas===

====Areas of 2-dimensional figures====
[[File:Triangle_GeometryArea.svg|thumb|Triangle area <math>A=\tfrac{b\cdot h}{2}</math>]]
* A [[triangle]]: <math>\tfrac12Bh</math> (where ''B'' is any side, and ''h'' is the distance from the line on which ''B'' lies to the other vertex of the triangle). This formula can be used if the height ''h'' is known. If the lengths of the three sides are known then ''[[Heron's formula]]'' can be used: <math>\sqrt{s(s-a)(s-b)(s-c)}</math> where ''a'', ''b'', ''c'' are the sides of the triangle, and <math>s = \tfrac12(a + b + c)</math> is half of its perimeter.<ref name=AF/> If an angle and its two included sides are given, the area is <math>\tfrac12 a b \sin(C)</math> where {{math|''C''}} is the given angle and {{math|''a''}} and {{math|''b''}} are its included sides.<ref name=AF/> If the triangle is graphed on a coordinate plane, a matrix can be used and is simplified to the absolute value of <math>\tfrac12(x_1 y_2 + x_2 y_3 + x_3 y_1 - x_2 y_1 - x_3 y_2 - x_1 y_3)</math>. This formula is also known as the [[shoelace formula]] and is an easy way to solve for the area of a coordinate triangle by substituting the 3 points ''(x<sub>1</sub>,y<sub>1</sub>)'', ''(x<sub>2</sub>,y<sub>2</sub>)'', and ''(x<sub>3</sub>,y<sub>3</sub>)''. The shoelace formula can also be used to find the areas of other polygons when their vertices are known. Another approach for a coordinate triangle is to use [[calculus]] to find the area.
* A [[simple polygon]] constructed on a grid of equal-distanced points (i.e., points with [[integer]] coordinates) such that all the polygon's vertices are grid points: <math>i + \frac{b}{2} - 1</math>, where ''i'' is the number of grid points inside the polygon and ''b'' is the number of boundary points. This result is known as [[Pick's theorem]].<ref name="Pick">{{cite journal|last=Trainin|first=J.|date=November 2007|title=An elementary proof of Pick's theorem|journal=[[Mathematical Gazette]]|volume=91|issue=522|pages=536–540|doi=10.1017/S0025557200182270|s2cid=124831432}}</ref>

====Area in calculus====
[[File:Integral as region under curve.svg|thumb|alt=A diagram showing the area between a given curve and the x-axis|Integration can be thought of as measuring the area under a curve, defined by ''f''(''x''), between two points (here ''a'' and ''b'').]]
[[File:Areabetweentwographs.svg|thumb|alt=A diagram showing the area between two functions|The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions]]
* The area between a positive-valued curve and the horizontal axis, measured between two values ''a'' and ''b'' (b is defined as the larger of the two values) on the horizontal axis, is given by the integral from ''a'' to ''b'' of the function that represents the curve:<ref name=MathWorld/>
:<math> A = \int_a^{b} f(x) \, dx.</math>
* The area between the [[graph of a function|graphs]] of two functions is [[equality (mathematics)|equal]] to the [[integral]] of one [[function (mathematics)|function]], ''f''(''x''), [[subtraction|minus]] the integral of the other function, ''g''(''x''):
:<math> A = \int_a^{b}  ( f(x) - g(x) ) \, dx, </math> where <math> f(x) </math> is the curve with the greater y-value.
* An area bounded by a function <math>r = r(\theta)</math> expressed in [[polar coordinates]] is:<ref name=MathWorld/>
:<math>A = {1 \over 2} \int r^2 \, d\theta. </math>
* The area enclosed by a [[parametric curve]] <math>\vec u(t) = (x(t), y(t)) </math> with endpoints <math> \vec u(t_0) = \vec u(t_1) </math> is given by the [[line integral]]s:
::<math> \oint_{t_0}^{t_1} x \dot y \, dt  = - \oint_{t_0}^{t_1} y \dot x \, dt  =  {1 \over 2} \oint_{t_0}^{t_1} (x \dot y - y \dot x) \, dt </math>
: or the ''z''-component of
::<math>{1 \over 2} \oint_{t_0}^{t_1} \vec u \times \dot{\vec u} \, dt.</math>
:(For details, see {{slink|Green's theorem|Area calculation}}.) This is the principle of the [[planimeter]] mechanical device.

====Bounded area between two quadratic functions====
To find the bounded area between two [[quadratic function]]s, we first subtract one from the other, writing the difference as
<math display=block>f(x)-g(x)=ax^2+bx+c=a(x-\alpha)(x-\beta)</math>
where ''f''(''x'') is the quadratic upper bound and ''g''(''x'') is the quadratic lower bound. 
By the area integral formulas above and [[Vieta's formulas|Vieta's formula]], we can obtain that<ref>{{cite book|title=Matematika|url=https://books.google.com/books?id=NFkVfrZBqpUC&pg=PA51|publisher=PT Grafindo Media Pratama|isbn=978-979-758-477-1|pages=51–|url-status=live|archive-url=https://web.archive.org/web/20170320100900/https://books.google.com/books?id=NFkVfrZBqpUC&pg=PA51|archive-date=2017-03-20}}</ref><ref>{{cite book|title=Get Success UN +SPMB Matematika|url=https://books.google.com/books?id=uwqvITs8OaUC&pg=PA157|publisher=PT Grafindo Media Pratama|isbn=978-602-00-0090-9|pages=157–|url-status=live|archive-url=https://web.archive.org/web/20161223115304/https://books.google.com/books?id=uwqvITs8OaUC&pg=PA157|archive-date=2016-12-23}}</ref>
<math display=block>A=\frac{(b^2-4ac)^{3/2}}{6a^2}=\frac{a}{6}(\beta-\alpha)^3,\qquad a\neq0.</math>
The above remains valid if one of the bounding functions is linear instead of quadratic.

====Surface area of 3-dimensional figures====
* [[Cone]]:<ref name=MathWorldCone>{{cite web|url=http://mathworld.wolfram.com/Cone.html|title=Cone|publisher=[[Wolfram MathWorld]]|author-link=Eric W. Weisstein|author=Weisstein, Eric W.|access-date=6 July 2012|url-status=live|archive-url=https://web.archive.org/web/20120621230050/http://mathworld.wolfram.com/Cone.html|archive-date=21 June 2012}}</ref> <math>\pi r\left(r + \sqrt{r^2 + h^2}\right)</math>, where ''r'' is the radius of the circular base, and ''h'' is the height. That can also be rewritten as <math>\pi r^2 + \pi r l </math><ref name=MathWorldCone/> or <math>\pi r (r + l) \,\!</math> where ''r'' is the radius and ''l'' is the slant height of the cone. <math>\pi r^2 </math> is the base area while <math>\pi r l </math> is the lateral surface area of the cone.<ref name=MathWorldCone/>
* [[Cube]]: <math>6s^2</math>, where ''s'' is the length of an edge.<ref name=MathWorldSurfaceArea/>
* [[Cylinder]]: <math>2\pi r(r + h)</math>, where ''r'' is the radius of a base and ''h'' is the height. The <math>2\pi r</math> can also be rewritten as <math>\pi d</math>, where ''d'' is the diameter.
* [[Prism (geometry)|Prism]]: <math>2B + Ph</math>, where ''B'' is the area of a base, ''P'' is the perimeter of a base, and ''h'' is the height of the prism.
* [[Pyramid (geometry)|pyramid]]: <math>B + \frac{PL}{2}</math>, where ''B'' is the area of the base, ''P'' is the perimeter of the base, and ''L'' is the length of the slant.
* [[Rectangular prism]]: <math>2 (\ell w + \ell  h + w h)</math>, where <math>\ell</math> is the length, ''w'' is the width, and ''h'' is the height.

====General formula for surface area====
The general formula for the surface area of the graph of a continuously differentiable function <math>z=f(x,y),</math> where <math>(x,y)\in D\subset\mathbb{R}^2</math> and <math>D</math> is a region in the xy-plane with the smooth boundary:
: <math> A=\iint_D\sqrt{\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2+1}\,dx\,dy. </math>
An even more general formula for the area of the graph of a [[parametric surface]] in the vector form <math>\mathbf{r}=\mathbf{r}(u,v),</math> where <math>\mathbf{r}</math> is a continuously differentiable vector function of <math>(u,v)\in D\subset\mathbb{R}^2</math> is:<ref name="doCarmo"/>
: <math> A=\iint_D \left|\frac{\partial\mathbf{r}}{\partial u}\times\frac{\partial\mathbf{r}}{\partial v}\right|\,du\,dv. </math>

===List of formulas===

{| class="wikitable"
|+ Additional common formulas for area:
! Shape
! Formula
! Variables
|-
|[[Square]]
|<math>A=s^2</math>
|[[File:Simple square with sides marked.svg|120px]]
|-
|[[Rectangle]]
|<math>A=ab</math>
|[[File:Rechteck-ab.svg|120px]]
|-
|[[Triangle]]
|<math>A=\frac12bh \,\!</math>
|[[File:Dreieck-allg-bh.svg|100px]]
|-
|[[Triangle]]
|<math>A=\frac12 a b \sin(\gamma)\,\!</math>
|[[File:Dreieck-allg-w.svg|100px]]
|-
|[[Triangle]]<br>
([[Heron's formula]])
|<math>A=\sqrt{s(s-a)(s-b)(s-c)}\,\!</math>
|[[File:Dreieck-allg.svg|100px]] <math> s =\tfrac 1 2 (a+b+c)</math> 
|-
|[[Isosceles triangle]]
|<math>A=\frac{c}{4}\sqrt{4a^2-c^2}</math>
|[[File:Dreieck-gsch.svg|80px]]
|-
|Regular [[triangle]]<br>
([[equilateral triangle]])
||<math>A=\frac{\sqrt{3}}{4}a^2\,\!</math>
||[[File:Dreieck-gseit.svg|100px]]
|-
|[[Rhombus]]/[[Kite (geometry)|Kite]]
|<math>A=\frac12de</math>
|[[File:Raute-de.svg|160px]]
|-
|[[Parallelogram]]
|<math>A=ah_a\,\!</math>
|[[File:Parallelog-aha.svg|160px]]
|-
|[[Trapezoid]]
|<math>A=\frac{(a+c)h}{2} \,\!</math>
|[[File:Trapez-abcdh.svg|150px]]
|-
|Regular [[hexagon]]
|<math>A=\frac{3}{2} \sqrt{3}a^2\,\!</math>
|[[File:Hexagon-a.svg|100px]]
|-
|Regular [[octagon]]
|<math>A=2(1+\sqrt{2})a^2\,\!</math>
|[[File:Oktagon-a.svg|120px]]
|-
|[[Regular polygon]]<br>
(<math>n</math> sides)
|<math>A=n\frac{ar}{2}=\frac{pr}{2}</math><br>
<math>\quad =\tfrac 1 4 na^2\cot(\tfrac \pi n)</math><br>
<math>\quad = nr^2 \tan(\tfrac \pi n)</math><br>
<math>\quad =\tfrac{1}{4n}p^2\cot(\tfrac \pi n)</math><br>
<math>\quad =\tfrac{1}{2}nR^2 \sin(\tfrac{2\pi}{n}) \,\!</math>
|[[File:Oktagon-a-r-R.svg|150px|left]]
<math>p=na\ </math> ([[perimeter]])<br>
<math>r=\tfrac a 2 \cot(\tfrac \pi n),</math><br>
<math>\tfrac a 2= r\tan(\tfrac \pi n)=R\sin(\tfrac \pi n)</math><br>
<math>r:</math> [[incircle]] radius<br>
<math>R:</math> [[circumcircle]] radius
|-
|[[Circle]]
|<math>A=\pi r^2=\frac{\pi d^2}{4}</math><br>
(<math> d=2r: </math> [[diameter]])
|[[fILE:Kreis-r-tab.svg|100px]]
|-
|[[Circular sector]]
|<math>A=\frac{\theta}{2}r^2=\frac{L \cdot r}{2}\,\!</math>
|[[File:Circle arc.svg|120px]]
|-
|[[Ellipse]]
|<math>A=\pi ab \,\!</math>
|[[File:Ellipse-ab-tab.svg|120px]]
|-
|[[Integral]]
| <math>A=\int_a^b f(x)\mathrm{d}x ,\ f(x)\ge 0</math>
| [[File:Vase-f-fx-tab.svg|hochkant=0.2]]
|- 
|
|'''[[Surface area]]'''
|
|-
| [[Sphere (geometry)|Sphere]]<br />
| <math>A = 4\pi r^2 = \pi d^2</math>
| [[File:Kugel-1-tab.svg|100px]]
|-
| [[Cuboid]]
| <math>A = 2(ab+ac+bc)</math>
| [[File:Quader-1-tab.svg|150px]]
|-
|  [[Cylinder (geometry)|Cylinder]]<br>
(incl. bottom and top)
| <math>A = 2 \pi r (r + h)</math>
| [[File:Zylinder-1-tab.svg|120px]]
|-
| [[Cone]]<br>
(incl. bottom)
| <math>A = \pi r (r + \sqrt{r^2+h^2})</math>
| [[File:Kegel-1-tab.svg|120px]]
|-
| [[Torus]]
| <math>A = 4\pi^2 \cdot R \cdot r</math>
| [[File:Torus-1-tab.svg|200px]]
|-
| [[Surface of revolution]]
|  <math>A = 2\pi\int_a^b\! f(x)\sqrt{1+\left[f'(x)\right]^2}\mathrm{d}x</math><br>
(rotation around the  x-axis)
| [[File:Vase-1-tab.svg|220px]]
|-
|}

The above calculations show how to find the areas of many common [[shapes]].

The areas of irregular (and thus arbitrary) polygons can be calculated using the "[[Surveyor's formula]]" (shoelace formula).<ref name=Surveyor>{{cite journal|last1=Braden|first1=Bart|date=September 1986|title=The Surveyor's Area Formula|journal=The College Mathematics Journal|volume=17|issue=4|pages=326–337|doi=10.2307/2686282|url=http://www.maa.org/pubs/Calc_articles/ma063.pdf|access-date=15 July 2012|url-status=live|archive-url=https://web.archive.org/web/20120627180152/http://www.maa.org/pubs/Calc_articles/ma063.pdf|archive-date=27 June 2012|jstor=2686282}}</ref>

===Relation of area to perimeter===

The [[isoperimetric inequality]] states that, for a closed curve of length ''L'' (so the region it encloses has [[perimeter]] ''L'') and for area ''A'' of the region that it encloses,

:<math>4\pi A \le L^2,</math>

and  equality holds if and only if the curve is a [[circle]]. Thus a circle has the largest area of any closed figure with a given perimeter.

At the other extreme, a figure with given perimeter ''L'' could have an arbitrarily small area, as illustrated by a [[rhombus]] that is "tipped over" arbitrarily far so that two of its [[angle]]s are arbitrarily close to 0° and the other two are arbitrarily close to 180°.

For a circle, the ratio of the area to the [[circumference]] (the term for the perimeter of a circle) equals half the [[radius]] ''r''. This can be seen from the area formula ''πr''<sup>2</sup> and the circumference formula 2''πr''.

The area of a [[regular polygon]] is half its perimeter times the [[apothem]] (where the apothem is the distance from the center to the nearest point on any side).

===Fractals===

Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). But if the one-dimensional lengths of a  [[fractal]] drawn in two dimensions are all doubled, the spatial content of the fractal scales by a power of two that is not necessarily an integer. This power is called the [[fractal dimension]] of the fractal.
<ref name="Mandelbrot1983">
{{cite book
 |last        = Mandelbrot
 |first       = Benoît B.
 |title       = The fractal geometry of nature <!--Isn't valid anymore: https://books.google.com/books?id=0R2LkE3N7-oC-->
 |url         = https://books.google.com/books?id=JFX9mQEACAAJ
 |access-date  = 1 February 2012
 |year        = 1983
 |publisher   = Macmillan
 |isbn        = 978-0-7167-1186-5
 |url-status     = live
 |archive-url  = https://web.archive.org/web/20170320115652/https://books.google.com/books?id=JFX9mQEACAAJ
 |archive-date = 20 March 2017
}}</ref>