Area Light Sources

Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''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 (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).

The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (written as m²), 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 boundary, calculus is usually required to compute the area. Indeed, 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. do Carmo, Manfredo (1976). ''Differential Geometry of Curves and Surfaces''. Prentice-Hall. p. 98, In analysis, the area of a subset of the plane is defined using Lebesgue measure,Walter Rudin (1966). ''Real and Complex Analysis'', McGraw-Hill, . though not every subset is measurable. 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 (termed measurable sets) 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 (m²), square centimetres (cm²), square millimetres (mm²), square kilometres (km²), square feet (ft²), square yards (yd²), square miles (mi²), 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 (say length of 3 metres and width of 2 metres) 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.
:1 foot = 12 inches,
the relationship between square feet and square inches is
:1 square foot = 144 square inches,
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: Chapter 5.
* 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 India,
* 20 dhurki = 1 dhur
* 20 dhur = 1 khatha
* 20 khata = 1 bigha
* 32 khata = 1 acre

History

Circle area

In the 5th century BCE, Hippocrates of Chios was the first to show that the area of a disk (the region enclosed by a circle) 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''. (The circumference is 2''r'', and the area of a triangle is half the base times the height, yielding the area ''r''² for the disk.) Archimedes approximated the value of π (and hence the area of a unit-radius circle) 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 (and did the same with circumscribed polygons).

Swiss scientist Johann Heinrich Lambert in 1761 proved that π, the ratio of a circle's area to its squared radius, is irrational, meaning it is not equal to the quotient of any two whole numbers. English translation by Catriona and David Lischka. In 1794, French mathematician Adrien-Marie Legendre proved that π² is irrational; this also proves that π is irrational. In 1882, German mathematician Ferdinand von Lindemann proved that π is transcendental (not the solution of any polynomial equation with rational coefficients), confirming a conjecture made by both Legendre and Euler.

Triangle area

Heron (or Hero) of Alexandria found what is known as Heron's formula for the area of a triangle in terms of its sides, and a proof can be found in his book, ''Metrica'', written around 60 CE. It has been suggested that Archimedes knew the formula over two centuries earlier, and since ''Metrica'' is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work.

In 499 Aryabhata, a great mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, expressed the area of a triangle as one-half the base times the height in the ''Aryabhatiya'' (section 2.6).

A formula equivalent to Heron's was discovered by the Chinese independently of the Greeks. It was published in 1247 in ''Shushu Jiuzhang'' ("Mathematical Treatise in Nine Sections"), written by Qin Jiushao.

Quadrilateral area

In the 7th century CE,