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.
Other articles related to "area":
... and conquered the Baiyue territories, and many Han people began settling in the Lingnan area ... This migration led to the Chinese language being spoken in the Lingnan area ... over the Nanyue region, many Han people entered the area and lived together with the Nanyue population, consequently affecting the lifestyle of the Nanyue people as well as stimulating the spread ...
... Designated Market Area, a region of the country in which radio and television stations in the major city of the area are seen in homes and households, as ... The IOC code for Dominica Dublin Metropolitan Area, an Irish court and police jurisdiction ...
... The Zhengdong New Area, also known as Zhengzhou Eastern New District, similar to Hangzhou Bay New Area in Ningbo and Hengqin New Area in Zhuhai, is just one of dozens of major economic zones that ... established and developed Zhengdong New Area, Mr ... overall planning scheme for Zhengdong New Area ...
... telecommunications, a Wide Area Telephone Service (WATS) is a long distance service offering for customer dial-type telecommunications between a given customer station ... are issued a toll-free telephone number, typically beginning with a designated toll-free area code ... The first inward WATS area code issued was 800, with 888, 877, and 866 area codes being planned and implemented in the 1990s, The 855 code was ...
... Immediately south of that area, Americans built several fur trading outposts in succeeding years, including Fort Lisa in 1812 Fort Atkinson in 1819 Cabanné's Trading Post ... built a town called Cutler's Park in the area in 1846 ... The treaty and cession involving the Omaha area occurred in 1854 when the Omaha Tribe ceded most of east-central Nebraska ...
Famous quotes containing the word area:
“Whether we regard the Womens Liberation movement as a serious threat, a passing convulsion, or a fashionable idiocy, it is a movement that mounts an attack on practically everything that women value today and introduces the language and sentiments of political confrontation into the area of personal relationships.”
—Arianna Stassinopoulos (b. 1950)
“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 reinsmother or child? And the child has most of the ammunition!”
—Dorothy Corkville Briggs (20th century)
“... nothing is more human than substituting the quantity of words and actions for their character. But using imprecise words is very similar to using lots of words, for the more imprecise a word is, the greater the area it covers.”
—Robert Musil (18801942)