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:


 * area (a) = 100 square meters (m²)
 * hectare (ha) = 100 ares (a) = 10000 square meters (m²)
 * square kilometre (km²) = 100 hectars (ha) = 10000 ares (a) = 1000000 square metres (m²)
 * square megametre (Mm²) = 1012 square metres
 * square foot = 144 square inches = 0.09290304 square metres (m²)
 * square yard = 9 sqft = 0.83612736 square metres (m²)
 * square perch = 30.25 square yards = 25.2928526 square metres (m²)
 * 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 (m²)
 * square mile = 640 acre = 2.5899881103 square kilometers (km²)

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: $$\frac{Bh}{2}$$ (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 $$s = \frac{a + b + c}{2}$$ 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.

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.