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]

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

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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: $\backslash tfrac12Bh$ (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, 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: $i\; +\; \backslash frac\{b\}\{2\}\; -\; 1$, 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

[[File:|thumb|287px|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 $\{1\; \backslash over\; 2\}\; \backslash int\_0^\{2\backslash pi\}\; r^2\; \backslash ,\; d\backslash theta$.
• 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 $(u,v)\backslash in\; D\backslash subset\backslash mathbb\{R\}^2$:

$A=\backslash iint\_D\; \backslash left|\backslash frac\{\backslash partial\backslash mathbf\{r\}\}\{\backslash partial\; u\}\backslash times\backslash frac\{\backslash partial\backslash mathbf\{r\}\}\{\backslash partial\; v\}\backslash right|\backslash ,du\backslash ,dv.$^[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.

See also

• Equi-areal mapping
• Integral
• Orders of magnitude (area)—A list of areas by size.
• Perimeter
• Volume

References

Notes

External links

• Weisstein, Eric W., "Area" from MathWorld.
• Area formulas
• Conversion cable diameter to circle cross-sectional area and vice versa

Area is the amount of space a two dimensional (flat) surface takes up. It is useful because it is how much of a material is needed to make a hollow container; for example, how much wood is needed to make a wardrobe.

You can use different formulas to find the area of different shapes.

• Area of a rectangle is the length of any two touching sides multiplied together. In other words, length times width.
• Area of a triangle is half of the base multiplied by the perpendicular height. This can be found using the trigonometry formula, $A\; =\; 1/2\; ab\; \backslash sin\; c.$.
• Area of a circle: $A\; =\; \backslash pi\; r^2$.

The area of a flat object is related to the surface area and volume of a three-dimensional object.

The area under a curve can be found using integration, from calculus.

Some units used to measure area are square mile and square kilometre.