Editing Area (section)

===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>