Area
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{{Mergefrom|Area (geometry)|date=February 2008}}
{{otheruses1|the physical quantity}}
'''Area''' is a [[quantity]] expressing the two-[[dimension]]al size of a defined part of a [[surface]], typically a region bounded by a closed [[curve]]. The term '''surface area''' refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a [[polyhedron]].
==Units==
Units for '''measuring surface area''' include:
:[[Metric]]
::[[square metre]] (m²) = [[SI derived unit]]
::[[are]] (a) = 100 square metres (m²)
::[[hectare]] (ha) = 10,000 square metres (m²)
::[[square kilometre]] (km²) = 1,000,000 square metres (m²)
::[[square megametre]] (Mm²) = 10<sup>12</sup> square metres
::[[square foot]] = 144 square inches = 0.09290304 square metres (m²)
::[[square yard]] = 9 square feet = 0.83612736 square metres (m²)
::square [[perch (unit)|perch]] = 30.25 square yards = 25.2928526 square metres (m²)
::[[acre]] = 160 square perches or 4,840 square yards or 43,560 square feet = 4046.8564224 square metres (m²)
::[[square mile]] = 640 acres = 2.5899881103 square kilometres (km²)
==Useful formulas==
[[Image:Area.svg|right|200px]]
{| class="prettytable"s
|+ Common [[equation]]s for area:
! Shape
! Equation
! Variables
|-
|[[Square (geometry)|Square]]
|<math>s^2\,\!</math>
|<math>s</math> is the length of the side of the square.
|-
|Regular [[triangle]]
||<math>\frac{\sqrt{3}}{4}s^2\,\!</math>
||<math>s</math> is the length of one side of the triangle.
|-
|Regular [[hexagon]]
|<math>\frac{3\sqrt{3}}{2}s^2\,\!</math>
|<math>s</math> is the length of one side of the hexagon.
|-
|Regular [[octagon]]
|<math>2(1+\sqrt{2})s^2\,\!</math>
|<math>s</math> is the length of one side of the octagon.
|-
|Any regular polygon
|<math>\frac{1}{2}a p \,\!</math>
|<math>a</math> is the [[apothem]], or the radius of an inscribed circle in the polygon, and <math>p</math> is the perimeter of the polygon.
|-
|Any regular polygon
|<math>\frac{P^2/n} {4 \cdot \tan(\pi/n)}\,\!</math>
|<math> P</math> is the Perimeter and <math>n</math> is the number of sides.
|-
|Any regular polygon (using degree measure)
|<math>\frac{P^2/n} {4 \cdot \tan(180^\circ/n)}\,\! </math>
|<math> P </math> is the Perimeter and <math>n</math> is the number of sides.
|-
|[[Rectangle]]
|<math>lw \,\!</math>
|<math>l</math> and <math>w</math> are the lengths of the rectangle's sides (length and width).
|-
|[[Parallelogram]] (in general)
|<math>bh\,\!</math>
|<math>b</math> and <math>h</math> are the length of the base and the length of the perpendicular height, respectively.
|-
|[[Rhombus]]
|<math>\frac{1}{2}ab</math>
|<math>a</math> and <math>b</math> are the lengths of the two [[diagonals]] of the rhombus.
|-
|[[Triangle]]
|<math>\frac{1}{2}bh \,\!</math>
|<math>b</math> and <math>h</math> are the [[Base (geometry)|base]] and [[Altitude (triangle)|altitude]] (measured perpendicular to the base), respectively.
|-
|[[Triangle]]
|<math>\frac{1}{2} a b \sin C\,\!</math>
|<math>a</math> and <math>b</math> are any two sides, and <math>C</math> is the angle between them.
|-
|[[Circle]]
|<math>\pi r^2 ,\,\!</math> or <math>\pi d^2/4 \,\!</math>
|<math>r</math> is the radius and <math>d</math> the [[diameter]].
|-
|[[Ellipse]]
|<math>\pi ab \,\!</math>
|<math>a</math> and <math>b</math> are the [[semi-major axis|semi-major]] and [[semi-minor axis|semi-minor]] axes, respectively.
|-
|[[Trapezoid]]
|<math>\frac{1}{2}(a+b)h \,\!</math>
|<math>a</math> and <math>b</math> are the parallel sides and <math>h</math> the distance (height) between the parallels.
|-
|Total surface area of a [[Cylinder (geometry)|Cylinder]]
|<math>2\pi r^2+2\pi r h \,\!</math>
|<math>r</math> and <math>h</math> are the radius and height, respectively.
|-
|Lateral surface area of a cylinder
|<math>2 \pi r h \,\!</math>
|<math>r</math> and <math>h</math> are the radius and height, respectively.
|-
|Total surface area of a [[Cone (geometry)|Cone]]
|<math>\pi r (l + r) \,\!</math>
|<math>r</math> and <math>l</math> are the radius and [[slant height]], respectively.
|-
|Lateral surface area of a cone
|<math>\pi r l \,\!</math>
|<math>r</math> and <math>l</math> are the radius and slant height, respectively.
|-
|Total surface area of a [[sphere (geometry)|Sphere]]
|<math>4\pi r^2\,\!</math> or <math>\pi d^2\,\!</math>
|<math>r</math> and <math>d</math> are the radius and diameter, respectively.
|-
|Total surface area of an [[ellipsoid (geometry)|ellipsoid]]
|
|See the article.
|-
|[[Circular sector]]
|<math>\frac{1}{2} r^2 \theta \,\!</math>
|<math>r</math> and <math>\theta</math> are the radius and angle (in [[radians]]), respectively.
|-
|[[Square (geometry)|Square]] to circular area conversion
|<math>\frac{4}{\pi} A\,\!</math>
|<math>A</math> is the area of the [[square (geometry)|square]] in square units.
|-
|[[Circle|Circular]] to square area conversion
|<math>\frac{1}{4} C\pi\,\!</math>
|<math>C</math> is the area of the [[circle]] in circular units.
<!--
|-
|A revolution of f(x) about the x-axis
|<math>2 \pi \int_{a}^{b} |f(x)| \sqrt{1+(f'(x))^2}dx</math>
|-
|Area of surface of revolution of f(x) about the y-axis
|<math>2 \pi \int_{a}^{b} |x| \sqrt{1+(f'(x))^2}dx</math>
-->
|}
All of the above calculations show how to find the [[area]] of many [[shapes]].
The area of irregular polygons can be calculated using the "[[Surveyor's formula]]".<ref>[http://www.maa.org/pubs/Calc_articles/ma063.pdf]</ref>
==See also==
*[[Real matrices (2 x 2)#Equi-areal mapping|Equi-areal mapping]]
*[[Integral]]
*[[Orders of magnitude (area)]]—A list of areas by size.
*[[Volume]]
==External links==
{{Wiktionary}}
*[http://www.math.com/tables/geometry/areas.htm Area formulas]
*[http://www.sengpielaudio.com/calculator-cross-section.htm Conversion cable diameter to circle cross-sectional area and vice versa]
[[Category:Area|*]]
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