Area?qsrc=3044

This article is about the geometric quantity. For other uses, see Area (disambiguation).

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Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The surface area of a 3-dimensional solid is the total area of the exposed surface, such as the sum of the areas of the exposed sides of a polyhedron. Area is an important invariant in the differential geometry of surfaces.^[1]

Table of Contents

1	Units
2	Formulae
3	Additional formulae
	3.1	Areas of 2-dimensional figures
	3.2	Area in calculus
	3.3	Surface area of 3-dimensional figures
		3.3.1	General formula
4	Minimization
5	See also
6	References
	6.1	Notes
7	External links

Units

Units for measuring area, with exact conversions, include:

square metre (m²)
are (a) = 100 square metres (m²)
hectare (ha) = 100 ares = 10000 square metres
square kilometre (km²) = 100 hectares = 10000 ares = 1000000 square metres
square megametre (Mm²) = 1000000000000 square metres
square foot = 144 square inches = 0.09290304 square metres
square yard = 9 square feet = 0.83612736 square metres
square perch = 30.25 square yards = 25.2928526 square metres
acre = 10 square chains = one furlong by one chain = 160 square perches = 4840 square yards = 43560 square feet = 4046.8564224 square metres
square mile = 640 acres = 2.589988110336 square kilometres
Kanal (unit) = 0.125 acres.

Formulae

Common formulae for area:
Shape	Formula	Variables
Regular triangle (equilateral triangle)	$View formula on Wikipedia$	$s$ is the length of one side of the triangle.
Triangle	$View formula on Wikipedia$	$s$ is half the perimeter, $a$ , $b$ and $c$ are the length of each side.
Triangle	$View formula on Wikipedia$	$a$ and $b$ are any two sides, and $C$ is the angle between them.
Triangle	$View formula on Wikipedia$	$b$ and $h$ are the base and altitude (measured perpendicular to the base), respectively.
Square	$View formula on Wikipedia$	$s$ is the length of one side of the square.
Rectangle	$View formula on Wikipedia$	$l$ and $w$ are the lengths of the rectangle's sides (length and width).
Rhombus	$View formula on Wikipedia$	$a$ and $b$ are the lengths of the two diagonals of the rhombus.
Parallelogram	$View formula on Wikipedia$	$b$ is the length of the base and $h$ is the perpendicular height.
Trapezoid	$View formula on Wikipedia$	$a$ and $b$ are the parallel sides and $h$ the distance (height) between the parallels.
Regular hexagon	$View formula on Wikipedia$	$s$ is the length of one side of the hexagon.
Regular octagon	$View formula on Wikipedia$	$s$ is the length of one side of the octagon.
Regular polygon	$View formula on Wikipedia$	$s$ is the sidelength and $n$ is the number of sides.
Regular polygon	$View formula on Wikipedia$	$a$ is the apothem, or the radius of an inscribed circle in the polygon, and $p$ is the perimeter of the polygon.
Circle	$View formula on Wikipedia$	$r$ is the radius and $d$ the diameter.
Circular sector	$View formula on Wikipedia$	$r$ and $θ$ are the radius and angle (in radians), respectively.
Ellipse	$View formula on Wikipedia$	$a$ and $b$ are the semi-major and semi-minor axes, respectively.
Total surface area of a Cylinder	$View formula on Wikipedia$	$r$ and $h$ are the radius and height, respectively.
Lateral surface area of a cylinder	$View formula on Wikipedia$	$r$ and $h$ are the radius and height, respectively.
Total surface area of a Cone	$View formula on Wikipedia$	$r$ and $l$ are the radius and slant height, respectively.
Lateral surface area of a cone	$View formula on Wikipedia$	$r$ and $l$ are the radius and slant height, respectively.
Total surface area of a Sphere	$View formula on Wikipedia$	$r$ and $d$ are the radius and diameter, respectively.
Total surface area of an ellipsoid		See the article.
Total surface area of a Pyramid	$View formula on Wikipedia$	$B$ is the base area, $P$ is the base perimeter and $L$ is the slant height.
Square to circular area conversion	$View formula on Wikipedia$	$A$ is the area of the square in square units.
Circular to square area conversion	$View formula on Wikipedia$	$C$ is the area of the circle in circular units.

The above calculations show how to find the area of many common shapes.

The area of irregular polygons can be calculated using the "Surveyor's formula".^[2]

Additional formulae

Areas of 2-dimensional figures

a triangle: $View formula on Wikipedia$ (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: $View formula on Wikipedia$ (where a, b, c are the sides of the triangle, and $View formula on Wikipedia$ is half of its perimeter) If an angle and its two included sides are given, then area=absinC where C is the given angle and a and b are its included sides. If the triangle is graphed on a coordinate plane, a matrix can be used and is simplified to the absolute value of (x₁y₂+ x₂y₃+ x₃y₁ - x₂y₁- x₃y₂- x₁y₃) all divided by 2. 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, (x₁,y₁) (x₂,y₂) (x₃,y ₃). 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 Infinitesimal 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: $View formula on Wikipedia$ , 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.

Area in calculus

The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions

the area between the graphs of two functions is equal to the integral of one function, f(x), minus the integral of the other function, g(x).
an area bounded by a function r = r(θ) expressed in polar coordinates is $View formula on Wikipedia$ .
the area enclosed by a parametric curve $View formula on Wikipedia$ with endpoints $View formula on Wikipedia$ is given by the line integrals

View formula on Wikipedia

(see Green's theorem)

or the z-component of

View formula on Wikipedia

Surface area of 3-dimensional figures

cube: $6 s 2$ , where s is the length of the top side
rectangular box: $View formula on Wikipedia$ the length divided by height
cone: $View formula on Wikipedia$ , where r is the radius of the circular base, and h is the height. That can also be rewritten as $π r 2 + π r l$ where r is the radius and l is the slant height of the cone. $π r 2$ is the base area while $π r l$ is the lateral surface area of the cone.
prism: 2 × Area of Base + Perimeter of Base × Height

General formula

The general formula for the surface area of the graph of a continuously differentiable function $z = f (x, y),$ where $View formula on Wikipedia$ and $D$ is a region in the xy-plane with the smooth boundary:

View formula on Wikipedia

Even more general formula for the area of the graph of a parametric surface in the vector form $View formula on Wikipedia$ where $View formula on Wikipedia$ is a continuously differentiable vector function of $View formula on Wikipedia$ :

View formula on Wikipedia

^[1]

Minimization

Given a wire contour, the surface of least area spanning ("filling") it is a minimal surface. Familiar examples include soap bubbles.

The question of the filling area of the Riemannian circle remains open.

References

Notes

↑ ^1.0 ^1.1 do Carmo, Manfredo. Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976. Page 98.
↑ http://www.maa.org/pubs/Calc_articles/ma063.pdf

External links

Look up area in Wiktionary, the free dictionary.

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