Area

Area is the measure of a region's size on a surface. The area of a plane region or plane area refers to the area of a shape or planar lamina, while the surface area refers to the area of an open surface or the boundary of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analogue of the length of a curve or the volume of a solid.
Two different regions may have the same area ; by synecdoche, "area" sometimes is used to refer to the region, as in a "polygonal area".
The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units, the standard unit of area is the square metre, which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.
There are several well-known formulas for the areas of simple shapes such as triangles, rectangles, and circles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. For shapes with curved boundaries, calculus is usually required to compute the area. In all, the problem of determining the area of plane figures was a major motivation for the historical development of calculus.
For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area. Formulas for the surface areas of simple shapes were computed by the ancient Greeks, but computing the surface area of a more complicated shape usually requires multivariable calculus.
Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, though not every subset is measurable if one supposes the axiom of choice. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.
Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists.

Formal definition

An approach to defining what is meant by "area" is through axioms. "Area" can be defined as a function from a collection M of a special kinds of plane figures to the set of real numbers, which satisfies the following properties:

For all S in M,.
If S and T are in M then so are and, and also.
If S and T are in M with then is in M and.
If a set S is in M and S is congruent to T then T is also in M and.
Every rectangle R is in M. If the rectangle has length h and breadth k then.
Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i.e.. If there is a unique number c such that for all such step regions S and T, then.

It can be proved that such an area function actually exists.

Units

Every unit of length has a corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measured in square metres, square centimetres, square millimetres, square kilometres, square feet, square yards, square miles, and so forth. Algebraically, these units can be thought of as the squares of the corresponding length units.
The SI unit of area is the square metre, which is considered an SI derived unit.

Conversions

Calculation of the area of a square whose length and width are 1 metre would be:
1 metre × 1 metre = 1 m²
and so, a rectangle with different sides would have an area in square units that can be calculated as:
3 metres × 2 metres = 6 m². This is equivalent to 6 million square millimetres. Other useful conversions are:

1 square kilometre = 1,000,000 square metres
1 square metre = 10,000 square centimetres = 1,000,000 square millimetres
1 square centimetre = 100 square millimetres.
Non-metric units

In non-metric units, the conversion between two square units is the square of the conversion between the corresponding length units.
the relationship between square feet and square inches is
where 144 = 12² = 12 × 12. Similarly:

1 square yard = 9 square feet
1 square mile = 3,097,600 square yards = 27,878,400 square feet

In addition, conversion factors include:

1 square inch = 6.4516 square centimetres
1 square foot = square metres
1 square yard = square metres
1 square mile = square kilometres
Other units including historical

There are several other common units for area. The are was the original unit of area in the metric system, with:

1 are = 100 square metres

Though the are has fallen out of use, the hectare is still commonly used to measure land:

1 hectare = 100 ares = 10,000 square metres = 0.01 square kilometres

Other uncommon metric units of area include the tetrad, the hectad, and the myriad.
The acre is also commonly used to measure land areas, where

1 acre = 4,840 square yards = 43,560 square feet.

An acre is approximately 40% of a hectare.
On the atomic scale, area is measured in units of barns, such that:

1 barn = 10⁻²⁸ square meters.

The barn is commonly used in describing the cross-sectional area of interaction in nuclear physics.
In South Asia, although the countries use SI units as official, many South Asians still use traditional units. Each administrative division has its own area unit, some of them have same names, but with different values. There's no official consensus about the traditional units values. Thus, the conversions between the SI units and the traditional units may have different results, depending on what reference that has been used.
Some traditional South Asian units that have fixed value:

1 Killa = 1 acre
1 Ghumaon = 1 acre
1 Kanal = 0.125 acre
1 Decimal = 48.4 square yards
1 Chatak = 180 square feet
History

Circle area

In the 5th century BCE, Hippocrates of Chios was the first to show that the area of a disk is proportional to the square of its diameter, as part of his quadrature of the lune of Hippocrates, but did not identify the constant of proportionality. Eudoxus of Cnidus, also in the 5th century BCE, also found that the area of a disk is proportional to its radius squared.
Subsequently, Book I of Euclid's Elements dealt with equality of areas between two-dimensional figures. The mathematician Archimedes used the tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, in his book Measurement of a Circle. Archimedes approximated the value of with his doubling method, in which he inscribed a regular triangle in a circle and noted its area, then doubled the number of sides to give a regular hexagon, then repeatedly doubled the number of sides as the polygon's area got closer and closer to that of the circle.

Triangle area

Quadrilateral area

In the 7th century CE, Brahmagupta developed a formula, now known as Brahmagupta's formula, for the area of a cyclic quadrilateral in terms of its sides. In 1842, the German mathematicians Carl Anton Bretschneider and Karl Georg Christian von Staudt independently found a formula, known as Bretschneider's formula, for the area of any quadrilateral.

General polygon area

The development of Cartesian coordinates by René Descartes in the 17th century allowed the development of the surveyor's formula for the area of any polygon with known vertex locations by Gauss in the 19th century.

Areas determined using calculus

The development of integral calculus in the late 17th century provided tools that could subsequently be used for computing more complicated areas, such as the area of an ellipse and the surface areas of various curved three-dimensional objects.

Area formulas

Polygon formulas

For a non-self-intersecting polygon, the Cartesian coordinates of whose n vertices are known, the area is given by the surveyor's formula:
where when i=''n-1, then i''+1 is expressed as modulus n and so refers to 0.

Rectangles

The most basic area formula is the formula for the area of a rectangle. Given a rectangle with length and width, the formula for the area is:
That is, the area of the rectangle is the length multiplied by the width. As a special case, as in the case of a square, the area of a square with side length is given by the formula:
The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a definition or axiom. On the other hand, if geometry is developed before arithmetic, this formula can be used to define multiplication of real numbers.

Dissection, parallelograms, and triangles

Most other simple formulas for area follow from the method of dissection.
This involves cutting a shape into pieces, whose areas must sum to the area of the original shape.
For an example, any parallelogram can be subdivided into a trapezoid and a right triangle, as shown in figure to the left. If the triangle is moved to the other side of the trapezoid, then the resulting figure is a rectangle. It follows that the area of the parallelogram is the same as the area of the rectangle:
However, the same parallelogram can also be cut along a diagonal into two congruent triangles, as shown in the figure to the right. It follows that the area of each triangle is half the area of the parallelogram:
Similar arguments can be used to find area formulas for the trapezoid as well as more complicated polygons.