Area

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 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. Area is an important invariant in the differential geometry of surfaces.

Units
Units for measuring area include:


 * are (a) = 100 square metres (m²)
 * hectare (ha) = 100 ares (a) = 10000 square metres
 * square kilometre (km²) = 100 hectares (ha) = 10000 ares = 1000000 square metres
 * square megametre (Mm²) = 1012 square metres
 * square foot = 144 square inches = 0.09290304 square metres
 * square yard = 9 sqft = 0.83612736 square metres
 * square perch = 30.25 square yards = 25.2928526 square metres
 * acre = 10 square chains (also one furlong by one chain); or 160 square perches; or 4840 square yards; or 43560 sqft = 4046.8564224 square metres
 * square mile = 640 acre = 2.5899881103 square kilometers

Formulæ


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".

Areas of 2-dimensional figures

 * a triangle: $$\tfrac12Bh$$ (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: $$\sqrt{s(s-a)(s-b)(s-c)}$$(where a, b, c are the sides of the triangle, and $$s = \tfrac12(a + b + c)$$ 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: $$i + \frac{b}{2} - 1$$, 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 $$ {1 \over 2} \int_0^{2\pi} r^2 \, d\theta $$.
 * the area enclosed by a parametric curve $$\vec u(t) = (x(t), y(t)) $$ with endpoints $$ \vec u(t_0) = \vec u(t_1) $$ is given by the line integrals
 * $$ \oint_{t_0}^{t_1} x \dot y \, dt = - \oint_{t_0}^{t_1} y \dot x \, dt  =  {1 \over 2} \oint_{t_0}^{t_1} (x \dot y - y \dot x) \, dt $$

(see Green's theorem)
 * or the z-component of


 * $${1 \over 2} \oint_{t_0}^{t_1} \vec u \times \dot{\vec u} \, dt.$$

Surface area of 3-dimensional figures

 * cube: $$6s^2$$, where s is the length of the top side
 * rectangular box: $$2 (\ell w + \ell h + w h)$$ the length divided by height
 * cone: $$\pi r\left(r + \sqrt{r^2 + h^2}\right)$$, where r is the radius of the circular base, and h is the height. That can also be rewritten as $$\pi r^2 + \pi r l $$ where r is the radius and l is the slant height of the cone. $$\pi r^2 $$ is the base area while $$\pi r l $$ 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 $$(x,y)\in D\subset\mathbb{R}^2$$ and $$D$$ is a region in the xy-plane with the smooth boundary:
 * $$ A=\iint_D\sqrt{\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2+1}\,dx\,dy. $$

Even more general formula for the area of the graph of a parametric surface in the vector form $$\mathbf{r}=\mathbf{r}(u,v),$$ where $$\mathbf{r}$$ is a continuously differentiable vector function of $$(u,v)\in D\subset\mathbb{R}^2$$:
 * $$ A=\iint_D \left|\frac{\partial\mathbf{r}}{\partial u}\times\frac{\partial\mathbf{r}}{\partial v}\right|\,du\,dv. $$

Area minimisation
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.