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.do Carmo, Manfredo. Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976. Page 98.

Units

Units for measuring area include:

Formulæ

{| class="wikipedia
|+ Common formulæ for area:
! Shape
! Equation
! 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
|$\backslash tfrac12\; a\; b\; \backslash sin(C)\backslash ,\backslash !$
|$a$ 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
|$bh\backslash ,\backslash !$
|$b$ and $h$ are the length of the base and the length of the perpendicular height, respectively.
|-
|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 sidelength 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.
|-

|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 formulæ

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

• 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

(see Green's theorem)

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:

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 $(u,v)\backslash in\; D\backslash subset\backslash mathbb\{R\}^2$:

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.

See also

• Equi-areal mapping

• Integral

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

• Volume