# Difference between revisions of "Area"

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The combined area of these three shapes is approximately 15.57 squares.

Area is a quantity that expresses the extent of a two-dimensional surface or shape, or planar lamina, in the plane. 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.[1] It is the two-dimensional analog 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.[2] 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.[3] 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.[4] 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.[5]

For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area.[1][6] 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.[7] In analysis, the area of a subset of the plane is defined using Lebesgue measure,[8] though not every subset is measurable.[9] In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.[1]

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

{{#invoke:see also|seealso}} An approach to defining what is meant by "area" is through axioms. "Area" can be defined as a function from a collection M of special kind of plane figures (termed measurable sets) to the set of real numbers which satisfies the following properties:

• For all S in M, a(S) ≥ 0.
• If S and T are in M then so are ST and ST, and also a(ST) = a(S) + a(T) − a(ST).
• If S and T are in M with ST then TS is in M and a(TS) = a(T) − a(S).
• If a set S is in M and S is congruent to T then T is also in M and a(S) = a(T).
• Every rectangle R is in M. If the rectangle has length h and breadth k then a(R) = hk.
• 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. SQT. If there is a unique number c such that a(S) ≤ c ≤ a(T) for all such step regions S and T, then a(Q) = c.

It can be proved that such an area function actually exists.[10]

## 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.[11] 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.[3]

### Conversions

Although there are 10 mm in 1 cm, there are 100 mm2 in 1 cm2.

The conversion between two square units is the square of the conversion between the corresponding length units. For example, since

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 kilometer = 1,000,000 square meters
• 1 square meter = 10,000 square centimetres = 1,000,000 square millimetres
• 1 square centimetre = 100 square millimetres
• 1 square yard = 9 square feet
• 1 square mile = 3,097,600 square yards = 27,878,400 square feet

### Other units

{{#invoke:see also|seealso}} 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:[11]

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

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:[11]

• 1 barn = 10−28 square meters.

The barn is commonly used in describing the cross sectional area of interaction in nuclear physics.[11]

In India,

• 20 Dhurki = 1 Dhur
• 20 Dhur = 1 Khatha
• 20 Khata = 1 Bigha
• 32 Khata = 1 Acre

## Area formulae

### Polygon formulae

#### Rectangles

The area of this rectangle is Template:Mvar.

The most basic area formula is the formula for the area of a rectangle. Given a rectangle with length Template:Mvar and width Template:Mvar, the formula for the area is:[2]

Template:Bigmath (rectangle)

That is, the area of the rectangle is the length multiplied by the width. As a special case, as l = w in the case of a square, the area of a square with side length Template:Mvar is given by the formula:[1][2]

Template:Bigmath (square)

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.

Equal area figures.

#### Dissection formulae

Most other simple formulae 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:[2]

Template:Bigmath  (parallelogram).
Two equal triangles.

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:[2]

${\displaystyle A={\frac {1}{2}}bh}$  (triangle).

## References

1. Template:Cite web
2. Template:Cite web
3. Bureau International des Poids et Mesures Resolution 12 of the 11th meeting of the CGPM (1960), retrieved 15 July 2012
4. {{#invoke:citation/CS1|citation |CitationClass=book }}
5. {{#invoke:citation/CS1|citation |CitationClass=book }}
6. Template:Cite web
7. do Carmo, Manfredo. Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976. Page 98, ISBN 978-0-13-212589-5
8. Walter Rudin, Real and Complex Analysis, McGraw-Hill, 1966, ISBN 0-07-100276-6.
9. Gerald Folland, Real Analysis: modern techniques and their applications, John Wiley & Sons, Inc., 1999,Page 20,ISBN 0-471-31716-0
10. {{#invoke:citation/CS1|citation |CitationClass=book }}
11. {{#invoke:Citation/CS1|citation |CitationClass=journal }} Chapter 5.
12. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
13. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
14. Template:Cite web