Area
Area is the content of a two-dimensional manifold (a surface), corresponding to the length in one dimension, to volume in three dimensions, and to bulk in four. Roughly speaking, area can be thought of as the "size" of the manifold, or as the amount of "material" that it would take to cover the manifold completely.
While area is fundamentally a two-dimensional property, it is possible to define an area associated with a three-dimensional manifold (a cast). One important such area is the surface area, the area of the (two-dimensional) boundary of a cast. (The name is a bit misleading, perhaps, because all areas pertain to surfaces by definition.) Another is the cross sectional area, the area of the intersection of the manifold with a plane (or, more rarely, with some other two-dimensional shape). Unlike the surface area, the cross-sectional area of a three-dimensional manifold is not uniquely defined; it depends on the particular two-dimensional manifold the three-dimensional manifold is intersected with, as well as where and at what angle the intersection takes place. Often a planar cross section is taken where the area is largest, but this is not a universal standard.
Area is usually abbreviated A, though surface area is often abbreviated S instead.
Calculations
The area of a shape can be determined the same way as for other-dimensional contents. Known formulæ exist for some widely used shapes, while determining the area of an arbitrary shape for which no formula is known may require calculus.
Formulæ
There are some shapes for which the area can be calculated through relatively simple formulæ. The area of a circle of radius r is [math]\displaystyle{ \pi r^2 }[/math]; the area of an ellipse of radii a and b is simply [math]\displaystyle{ \pi a b }[/math] (of which formula the circle formula may be taken as a special case). Even more generally, the area of a superellipse of radii a and b and of exponent [math]\displaystyle{ n }[/math] is equal to [math]\displaystyle{ 4 a b \frac{(\Gamma ( 1 + \frac {1}{n} ) )^2}{\Gamma (1 + \frac {2}{n} ) } }[/math].
The area of a regular polygon of n sides each of length s is [math]\displaystyle{ \frac{1}{4} n s^2 \cot (\frac {\pi}{n}) }[/math]. The area of a rectangle is just the product of its length and width. As a special case of both formulae, the area of a square of side length s is [math]\displaystyle{ s^2 }[/math]. The area of a triangle is generally given as [math]\displaystyle{ \frac{1}{2} b h }[/math], with b and h respectively its base and height, but there are other forms of this formula for other information, such as Hero's formula when the lengths of the three sides of the triangle are known. The area of an arbitrary polygon, as long the coordinates of its vertices are known, can be derived through an expression sometimes called the shoelace formula.
The area of a lemniscate with a focal distance of [math]\displaystyle{ a }[/math] is simply [math]\displaystyle{ a^2 }[/math]. The area of the rose [math]\displaystyle{ r = a \sin n \theta }[/math], [math]\displaystyle{ n \in Z^+ }[/math], is equal to [math]\displaystyle{ \frac {\pi a^2}{2} }[/math] if [math]\displaystyle{ n }[/math] is even, and [math]\displaystyle{ \frac {\pi a^2}{4} }[/math] if [math]\displaystyle{ n }[/math] is odd. The area of a loopless limaçon [math]\displaystyle{ r = b + a \cos \theta }[/math] with [math]\displaystyle{ b \lt = a }[/math] is equal to [math]\displaystyle{ \frac{\pi}{2} ( a^2 + 2 b^2) }[/math]; when [math]\displaystyle{ b = a }[/math] this becomes a cardioid, and the area reduces to [math]\displaystyle{ \frac{3}{2} a^2 }[/math]. For a limaçon with a loop (when [math]\displaystyle{ b \gt a }[/math]), the area of the loop is [math]\displaystyle{ \frac{1}{2}( a^2 + 2 b^2 )\cos^{-1} \frac{b}{a} - \frac {3}{2} b \sqrt{a^2 - b^2} }[/math], and the area of the rest of the limaçon is [math]\displaystyle{ ( a^2 + 2 b^2)\sin^{-1} \frac{b}{a} + 3 b \sqrt{a^2 - b^2} }[/math].
While the above formulæ all deal with flat shapes, formulæ can of course be derived for areas of curved surfaces as well. The area of a spherical triangle on a sphere of radius R, for example, can be written as [math]\displaystyle{ A = R^2 (A + B + C - \pi) }[/math], where A, B, and C are the triangle's vertex angles in radians.
Integration
While formulæ exist for these special cases, the area of an arbitrary shape can be found through methods of calculus (which is in fact how most of these formulæ can be derived). In a flat two-dimensional plane, the area under a curve (that is, between the curve and the x axis) can be found by integrating the equation of the curve over the x coordinate. For example, if we wanted to find the area under the parabola [math]\displaystyle{ y = -x^2+4 }[/math] between [math]\displaystyle{ x=-2 }[/math] and [math]\displaystyle{ x=2 }[/math], we would integrate [math]\displaystyle{ A = \int_{2}^{2} (-x^2+4) dx = \left [ -\frac{1}{3}x^3 + 4x \right ]_-2^2 = \left (-\frac{8}{3} + 8 \right ) - \left (\frac{8}{3} - 8 \right ) = \frac{32}{3} }[/math]. The area between two curves [math]\displaystyle{ f(x) }[/math] and [math]\displaystyle{ g(x) }[/math] is simply the integral of the difference of the areas under each curve, [math]\displaystyle{ \int \left ( f(x) - g(x) \right ) dx }[/math].
As for contents of other dimensions, the area of a flat surface can be calculated by taking the double integral across the appropriate dimensions—often, appropriately, called an area integral. In Cartesian coordinates, this simply means [math]\displaystyle{ A = \iint\;dx\;dy }[/math]; in polar coordinates it becomes [math]\displaystyle{ A = \iint\;r\;dr\;d\theta }[/math]; in parabolic coordinates [math]\displaystyle{ A = \iint\;(u^2 + v^2)\;du\;dv }[/math]; and so on. For a non-flat surface, an area integral can still be used, but determining the infinitesimal area element may be less straightforward. If, for example, the surface in question can be expressed in the form [math]\displaystyle{ z(x,y) }[/math] (that is, if the z-coordinate can be expressed as a function of the x and y-coordinates), then the area of the surface is equal to [math]\displaystyle{ \iint {\sqrt { \left ( \frac{df}{dx} \right )^2 + \left ( \frac{df}{dy} \right)^2 + 1} \; dx\;dy} }[/math].
Limits
Even these methods may not work in all cases. For iteratively defined fractals, there may be no straightforward way of calculating the area of the final fractal, but it may be derived as the limit of the sequence of the areas at each step. For instance, consider a Koch snowflake, one of the best-known fractals, with a width of one unit. The iterative process of the snowflake's creation starts with an equilateral triangle of area [math]\displaystyle{ \frac{\sqrt{3}}{4} }[/math]. In the second step, three new triangles are added with 1/3 the edge length of the original triangle, and therefore 1/9 the area... but since there are three of them, the total area added is 1/3 that of the original triangle, making the total area added [math]\displaystyle{ \frac {1}{3} \left ( \frac{\sqrt{3}}{4} \right ) = \frac{\sqrt {3}}{12} }[/math], and the total area [math]\displaystyle{ \frac{\sqrt{3}}{4} + \frac{\sqrt {3}}{12} = \frac{4 \sqrt{3}}{4} = \frac{\sqrt{3}}{3} }[/math]. Now, in each step thereafter, four times as many triangles are added than the previous step, but with the area of each triangle only one ninth that of those of the previous step, so the total area added is 4/9 as much as the previous step. The area added in the nth iteration can therefore be expressed as [math]\displaystyle{ \frac{\sqrt{3}}{12} \left ( \frac{4}{9} \right )^{(n-1)} }[/math], and the total area after the nth iteration as [math]\displaystyle{ \frac {\sqrt{3}}{4} + \sum_{k=1}^n {\frac{\sqrt{3}}{12} \left ( \frac{4}{9} \right )^{(k-1)}} = \frac {\sqrt{3}}{4} + \frac{\sqrt{3}}{12} \sum_{k=0}^{n-1} {\left ( \frac{4}{9} \right )^k} }[/math]... a geometric series which converges at the limit to [math]\displaystyle{ \frac{\sqrt{3}}{4} + \frac{\sqrt{3}}{12} \sum_{k=0}^\infty {\left ( \frac{4}{9} \right )^k} = \frac{\sqrt{3}}{4} + \frac{\sqrt{3}}{12} \frac{9}{5} = \frac{\sqrt{3}}{4} + \frac{3 \sqrt{3}}{20} = \frac{2 \sqrt{3}}{5} }[/math].
Note that the area of a fractal need not be a finite, nonzero number. For instance, the Sierpiński gasket is produced by removing at each iteration inner triangles composing 1/4 of the area of the previous step, meaning that at each iteration the area is multipled by 3/4. This makes the area after n steps the initial area times [math]\displaystyle{ \left ( \frac{3}{4} \right )^n }[/math]... which at the limit as [math]\displaystyle{ n \to \infty }[/math] goes to zero... the Sierpiński gasket has a zero area. Meanwhile, while its three-dimensional analogue the tetrix has a finite and nonzero surface area—indeed, exactly the same as the surface area of the regular tetrahedron it starts with, since each iteration adds surfaces of the same total area as it removes, thus leaving the surface area unchanged—its orthohedral cousin the Menger sponge does not. The surface area of the Menger sponge is a little harder to calculate, but it turns out that after the nth iteration the surface area of a Menger sponge starting with a cube of side length s is [math]\displaystyle{ \frac{4 \cdot 8^n + 2 \cdot 20^n}{3^n} }[/math] which diverges as [math]\displaystyle{ n \to \infty }[/math]... the surface area of the Menger sponge is infinite.
The fractal dimension of a fractal can give some hints about its area. In general, fractals of dimension 2 have finite area; fractals of dimension between 1 and 2 have zero area (and infinite perimeter); fractals of dimension between 2 and 3 have infinite surface area.
Others
There are some cases for which no analytical method of calculating area is known, or perhaps possible. The area of the Mandelbrot set is one such case: it can certainly be approximated numerically, but there is (at the time of this writing) no known closed-form expression for its exact value. (There is a known expression for the area of the Mandelbrot set in terms of certain coefficients, but there is no known closed form for these coefficients—they can only be calculated recursively, and rather inefficiently at that—so this doesn't solve the problem.)
Units
Typically, area is just measured in units of length squared. For example, the standard metric unit of area is the square meter (or the square centimeter in cgs), though smaller areas may be measurable in square centimeters, square millimeters, and so on, or larger as square kilometers, square terameters, etc. In the English system area may be measured in square feet, square inches, square yards, square miles, and so forth, as appropriate. Effectively, the square of any unit of length can be a unit of area; one could (but probably wouldn't) measure areas in square parsecs, square cubits, or square smoots.
There are also some units of area that are defined directly and not as squares of units of length. One such metric unit is the are, defined as one hundred square meters, though to it's probably more familiar in the prefixed form of "hectare" (one hundred ares, or ten thousand square meters). An English unit of area still in common use in many places is the acre, equal to 43,560 square feet. (One quarter of an acre used to be called a rood, but this is nowadays seldom used.) The acre was originally defined as the amount of land a team of oxen can plow in a day... an important enough consideration in earlier days that many other cultures defined units of area similarly.
Some other, lesser known units of area include the following (units with an asterisk are defined in terms of area plowed, like the acre):
- Barn: A unit used in nuclear physics defined as 10-28 square meters (roughly the cross sectional area of a uranium nucleus)
- Bigha: A unit of land area used in India and some neighboring nations, its size varying regionally but generally equal to several thousand square meters. One twentieth of a bigha is sometimes called a katha. Other (similarly variable) units used in South Asia are the kanal (about 500 square meters), the karm (about three square meters) , the keela (about 4,000 square meters), the marla (about 20 square meters), and the sarsahi (a little over two square meters).
- Bunder: A Dutch and Belgian unit now defined as equal to one hectare.
- Dunam*: A unit of area formerly used in the Ottoman Empire, and still used in some areas that used to belong to it. While its former definition varied, it has in modern times been defined as equal to exactly one thousand square meters (one decare).
- Feddan (فدّاʆ)*: A unit of area used in various Arabic countries, equal to 4,200 square units. A feddan is divided into 24 kirats (قيراط).
- Hide: An English unit of area representing the minimum amount of land that can support a household—and which therefore varied by time and region, but was generally about 120 acres.
- Mǔ (畝/亩): A Chinese unit of area, now defined as 2/3 of a hectare. One thousandth of a mǔ is a háo (毫); one hundredth is a lí (釐/厘); one tenth is a fēn (市分); and ten mǔ are a qǐng (頃/顷).
- Ngaan (งาน): A unit of area commonly used in Thailand, equal to 400 square meters. Four ngaan compose a rai.
- Quinaria: A unit of area used in ancient Rome to measure the cross-sectional area of a pipe, equal to about 4.5 square centimeters.
- Stremma (στρέμμα)*: A unit of measurement originating in ancient Greece, still used but now standardized to exactly 1,000 square meters.
- Tsubo (坪): A Japanese unit of area defined as the area of two tatami mats, and standardized as equal to about 3.306 square meters. An equivalent unit used in agriculture is a bu (歩). Various fractions and multiples of the tsubo/bu have their own names: 1/100 of a tsubo is a shaku (勺), 1/10 is a gō (合), 1/2 a jō (畳); 30 bu are a se (畝), 300 a tan (段/反), and 3,000 a chō (町/町歩).