Area

is between 15 and 16 squares.]] Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).

The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (m²), which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

There are several well-known formulas for the areas of simple shapes such as triangles, rectangles, and circles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus.

For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area. Formulas for the surface areas of simple shapes were computed by the ancient Greeks, but computing the surface area of a more complicated shape usually requires multivariable calculus.

Area plays a important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, though not every subset is measurable. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.

Formal definition

An approach to defining what is meant by area is through axioms. For example, we may define area as a function a from a collection M of special kind of plane figures (termed measurable sets) to the set of real numbers which satisfies the following properties:

For all S in M, $a(S)\; \backslash ge\; 0$.

If S and T are in M then so are $S\backslash cup\; T$ and $S\backslash cap\; T$ and also, $a(S\backslash cup\; T)=a(S)+a(T)-a(S\backslash cap\; T)$.

If S and T are in M with $S\backslash subset\; T$ then T − S is in M and a(T − S) = a(T) − a(S).

If a set S is in M and S is congruent to T then T is also in M and a(S) = a(T).

Every rectangle R is in M. If the rectangle has length h and breadth k then a(R) = hk.

Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i.e. $S\backslash subset\; Q\backslash subset\; T$. If there is a unique number c such that $a(S)\backslash le\; c\backslash le\; a(T)$ for all such step regions S and T, then a(Q) = c.

It can be proved that such a area function actually exists. (See, for example, Elementary Geometry from an Advanced Standpoint by Edwin Moise.)

Units

made of PVC pipe.]] Every unit of length has a corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measure in square metres (m²), square centimetres (cm²), square millimetres (mm²), square kilometres (km²), square feet (ft²), square yards (yd²), square miles (mi²), and so forth. Algebraically, these units can be thought of as the squares of the corresponding length units.

The SI unit of area is the square metre, which is considered an SI derived unit.

Conversions

The conversion between two square units is the square of the conversion between the corresponding length units. For example, since :1 foot = 12 inches, the relationship between square feet and square inches is :1 square foot = 144 square inches, where 144 = 12² = 12 × 12. Similarly:

1 square kilometer = 1,000,000 square meters

1 square meter = 10,000 square centimetres = 1,000,000 square millimetres

1 square centimetre = 100 square millimetres

1 square yard = 9 square feet

1 square mile = 3,097,600 square yards = 27,878,400 square feet

In addition,

1 square inch = 6.4516 square centimetres

1 square foot = square metres 1 square yard = square metres 1 square mile = square kilometres

Other units

There are several other common units for area. The are was the original unit of area in the metric system, with

1 are = 100 square metres

Though the are has fallen out of use, the hectare is still commonly used to measure land:

1 hectare = 100 ares = 10,000 square metres = 0.01 square kilometres

Other uncommon metric units of area include the tetrad, the hectad, and the myriad.

The acre is also commonly used to measure land areas, where

1 acre = 4,840 square yards = 43,560 square feet.

An acre is approximately 40% of a hectare.

Basic area formulae

Rectangles

The most basic area formula is the formula for the area of a rectangle. Given a rectangle with length and , the formula for the area is :  (rectangle). That is, the area of the rectangle is the length multiplied by the width. As a special case, the area of a square with side length is given by the formula :  (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 numbers.

Dissection formulae

Most other simple formulae for area follow from the method of dissection. This involves cutting a shape into pieces, whose areas must sum to the area of the original shape.

For 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: :  (parallelogram). However, the same parallelogram can also be cut along a diagonal into two 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: :$A\; =\; \backslash frac\{1\}\{2\}bh$  (triangle). Similar arguments can be used to find area formulae for the trapezoid and the rhombus, as well as more complicated polygons.

Circles

s which rearrange to form an approximate parallelogram.]] The formula for the area of a circle is based on a similar method. Given a circle of radius , it is possible to partition the circle into sectors, as shown in the figure to the right. Each sector is approximately triangular in shape, and the sectors can be rearranged to form and approximate parallelogram. The height of this parallelogram is , and the width is half the circumference of the circle, or . Thus, the total area of the circle is , or : :  (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 of the areas of the approximate parallelograms is exactly , which is the area of the circle.

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: :$A\; \backslash ;=\backslash ;\; \backslash int\_\{-r\}^r\; 2\backslash sqrt\{r^2\; -\; x^2\}\backslash ,dx\; \backslash ;=\backslash ;\; \backslash pi\; r^2.$

Surface area

showed that the surface area and volume of a sphere is exactly 2/3 of the area and volume of the surrounding cylindrical surface.]] Most basic formulae for surface area can be obtained by cutting surfaces and flattening them out. For example, if the side surface of a cylinder (or any 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, 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: because the surface of 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 :  (sphere). where 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.

List of formulae

{| class="wikipedia |+ Common formulae for area: ! Shape ! Formula ! Variables |- |Regular triangle (equilateral triangle) ||$\backslash tfrac14\backslash sqrt\{3\}s^2\backslash ,\backslash !$ ||$s$ is the length of one side of the triangle. |- |Triangle |$\backslash sqrt\{s(s-a)(s-b)(s-c)\}\backslash ,\backslash !$ |$s$ is half the perimeter, $a$, $b$ and $c$ are the length of each side. |- |Triangle |$\backslash tfrac12\; a\; b\; \backslash sin(C)\backslash ,\backslash !$ |$a$ and $b$ are any two sides, and $C$ is the angle between them. |- |Triangle |$\backslash tfrac12bh\; \backslash ,\backslash !$ |$b$ and $h$ are the base and altitude (measured perpendicular to the base), respectively. |- |Square |$s^2\backslash ,\backslash !$ |$s$ is the length of one side of the square. |- |Rectangle |$lw\; \backslash ,\backslash !$ |$l$ and $w$ are the lengths of the rectangle's sides (length and width). |- |Rhombus |$\backslash tfrac12ab$ |$a$ and $b$ are the lengths of the two diagonals of the rhombus. |- |Parallelogram |$bh\backslash ,\backslash !$ |$b$ is the length of the base and $h$ is the perpendicular height. |- |Trapezoid |$\backslash tfrac12(a+b)h\; \backslash ,\backslash !$ |$a$ and $b$ are the parallel sides and $h$ the distance (height) between the parallels. |- |Regular hexagon |$\backslash tfrac32\backslash sqrt\{3\}s^2\backslash ,\backslash !$ |$s$ is the length of one side of the hexagon. |- |Regular octagon |$2\backslash left(1+\backslash sqrt\{2\}\backslash right)s^2\backslash ,\backslash !$ |$s$ is the length of one side of the octagon. |- | rowspan=2 | Regular polygon |$\backslash frac\{ns^2\}\; \{4\; \backslash cdot\; \backslash tan(\backslash pi/n)\}\backslash ,\backslash !$ |$s$ is the side length and $n$ is the number of sides. |- |$\backslash tfrac12a\; p\; \backslash ,\backslash !$ |$a$ is the apothem, or the radius of an inscribed circle in the polygon, and $p$ is the perimeter of the polygon. |- |Circle |$\backslash pi\; r^2\backslash \; \backslash text\{or\}\backslash \; \backslash frac\{\backslash pi\; d^2\}\{4\}\; \backslash ,\backslash !$ |$r$ is the radius and $d$ the diameter. |- |Circular sector |$\backslash tfrac12\; r^2\; \backslash theta\; \backslash ,\backslash !$ |$r$ and $\backslash theta$ are the radius and angle (in radians), respectively. |- |Ellipse |$\backslash pi\; ab\; \backslash ,\backslash !$ |$a$ and $b$ are the semi-major and semi-minor axes, respectively. |- |Total surface area of a Cylinder |$2\backslash pi\; r\; (r\; +\; h)\backslash ,\backslash !$ |$r$ and $h$ are the radius and height, respectively. |- |Lateral surface area of a cylinder |$2\; \backslash pi\; r\; h\; \backslash ,\backslash !$ |$r$ and $h$ are the radius and height, respectively. |- |Total surface area of a Cone |$\backslash pi\; r\; (r\; +\; l)\; \backslash ,\backslash !$ |$r$ and $l$ are the radius and slant height, respectively. |- |Lateral surface area of a cone |$\backslash pi\; r\; l\; \backslash ,\backslash !$ |$r$ and $l$ are the radius and slant height, respectively. |- |Total surface area of a Sphere |$4\backslash pi\; r^2\backslash \; \backslash text\{or\}\backslash \; \backslash pi\; d^2\backslash ,\backslash !$ |$r$ and $d$ are the radius and diameter, respectively. |- |Total surface area of an ellipsoid |  |See the article. |- |Total surface area of a Pyramid |$B+\backslash frac\{P\; L\}\{2\}\backslash ,\backslash !$ |$B$ is the base area, $P$ is the base perimeter and $L$ is the slant height. |- |Square to circular area conversion |$\backslash frac\{4\}\{\backslash pi\}\; A\backslash ,\backslash !$ |$A$ is the area of the square in square units. |- |Circular to square area conversion |$\backslash frac\{1\}\{4\}\; C\backslash pi\backslash ,\backslash !$ |$C$ 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".

Additional formulae

Areas of 2-dimensional figures

a triangle: $\backslash tfrac12Bh$ (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: $\backslash sqrt\{s(s-a)(s-b)(s-c)\}$ where a, b, c are the sides of the triangle, and $s\; =\; \backslash tfrac12(a\; +\; b\; +\; c)$ is half of its perimeter. If an angle and its two included sides are given, the area is $\backslash tfrac12\; a\; b\; \backslash sin(C)$ 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 $\backslash tfrac12(x\_1\; y\_2\; +\; x\_2\; y\_3\; +\; x\_3\; y\_1\; -\; x\_2\; y\_1\; -\; x\_3\; y\_2\; -\; x\_1\; y\_3)$. 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), and (x₃,y₃). 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: $i\; +\; \backslash frac\{b\}\{2\}\; -\; 1$, 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 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 $\{1\; \backslash over\; 2\}\; \backslash int\_0^\{2\backslash pi\}\; r^2\; \backslash ,\; d\backslash theta$.

the area enclosed by a parametric curve $\backslash vec\; u(t)\; =\; (x(t),\; y(t))$ with endpoints $\backslash vec\; u(t\_0)\; =\; \backslash vec\; u(t\_1)$ is given by the line integrals

::$\backslash oint\_\{t\_0\}^\{t\_1\}\; x\; \backslash dot\; y\; \backslash ,\; dt\; =\; -\; \backslash oint\_\{t\_0\}^\{t\_1\}\; y\; \backslash dot\; x\; \backslash ,\; dt\; =\; \{1\; \backslash over\; 2\}\; \backslash oint\_\{t\_0\}^\{t\_1\}\; (x\; \backslash dot\; y\; -\; y\; \backslash dot\; x)\; \backslash ,\; dt$ (see Green's theorem) :or the z-component of

::$\{1\; \backslash over\; 2\}\; \backslash oint\_\{t\_0\}^\{t\_1\}\; \backslash vec\; u\; \backslash times\; \backslash dot\{\backslash vec\; u\}\; \backslash ,\; dt.$

Surface area of 3-dimensional figures

cube: $6s^2$, where s is the length of the top side

rectangular box: $2\; (\backslash ell\; w\; +\; \backslash ell\; h\; +\; w\; h)$ the length divided by height

cone: $\backslash pi\; r\backslash left(r\; +\; \backslash sqrt\{r^2\; +\; h^2\}\backslash right)$, where r is the radius of the circular base, and h is the height. That can also be rewritten as $\backslash pi\; r^2\; +\; \backslash pi\; r\; l$ where r is the radius and l is the slant height of the cone. $\backslash pi\; r^2$ is the base area while $\backslash pi\; r\; l$ 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 $(x,y)\backslash in\; D\backslash subset\backslash mathbb\{R\}^2$ and $D$ is a region in the xy-plane with the smooth boundary: : $A=\backslash iint\_D\backslash sqrt\{\backslash left(\backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\}\backslash right)^2+\backslash left(\backslash frac\{\backslash partial\; f\}\{\backslash partial\; y\}\backslash right)^2+1\}\backslash ,dx\backslash ,dy.$ Even more general formula for the area of the graph of a parametric surface in the vector form $\backslash mathbf\{r\}=\backslash mathbf\{r\}(u,v),$ where $\backslash mathbf\{r\}$ is a continuously differentiable vector function of $(u,v)\backslash in\; D\backslash subset\backslash mathbb\{R\}^2$: : $A=\backslash iint\_D\; \backslash left|\backslash frac\{\backslash partial\backslash mathbf\{r\}\}\{\backslash partial\; u\}\backslash times\backslash frac\{\backslash partial\backslash mathbf\{r\}\}\{\backslash partial\; v\}\backslash right|\backslash ,du\backslash ,dv.$

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.

See also

Equi-areal mapping

Integral

Orders of magnitude (area)—A list of areas by size.

Perimeter

Volume

References

Notes

External links

Area formulas

Conversion cable diameter to circle cross-sectional area and vice versa

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