**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. 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. 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. In analysis, the area of a subset of the plane is defined using Lebesgue measure, 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.

Read more about Area: Formal Definition, Units, Optimization

### Famous quotes containing the word area:

“Prosperous farmers mean more employment, more prosperity for the workers and the business men of ... every industrial *area* in the whole country.”

—Franklin D. Roosevelt (1882–1945)

“The *area* [of toilet training] is one where a child really does possess the power to defy. Strong pressure leads to a powerful struggle. The issue then is not toilet training but who holds the reins—mother or child? And the child has most of the ammunition!”

—Dorothy Corkville Briggs (20th century)

“Many women are reluctant to allow men to enter their domain. They don’t want men to acquire skills in what has traditionally been their *area* of competence and one of their main sources of self-esteem. So while they complain about the male’s unwillingness to share in domestic duties, they continually push the male out when he moves too confidently into what has previously been their exclusive world.”

—Bettina Arndt (20th century)