Area

This article is about the geometric quantity. For other uses, see Area (disambiguation).

Area
The areas of this square and this disk are the same.
Common symbols
A or S
SI unitSquare metre [m2]
In SI base units1 m2
Dimension

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 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.[1] 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). Two different regions may have the same area (as in squaring the circle); 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.[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][7] 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.[8] In analysis, the area of a subset of the plane is defined using Lebesgue measure,[9] though not every subset is measurable if one supposes the axiom of choice.[10] 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.

  1. ^ a b c Weisstein, Eric W. "Area". Wolfram MathWorld. Archived from the original on 5 May 2012. Retrieved 3 July 2012.
  2. ^ Cite error: The named reference AF was invoked but never defined (see the help page).
  3. ^ "Resolution 12 of the 11th meeting of the CGPM (1960)". Bureau International des Poids et Mesures. Archived from the original on 2012-07-28. Retrieved 15 July 2012.
  4. ^ Mark de Berg; Marc van Kreveld; Mark Overmars; Otfried Schwarzkopf (2000). "Chapter 3: Polygon Triangulation". Computational Geometry (2nd revised ed.). Springer-Verlag. pp. 45–61. ISBN 978-3-540-65620-3.
  5. ^ Boyer, Carl B. (1959). A History of the Calculus and Its Conceptual Development. Dover. ISBN 978-0-486-60509-8.
  6. ^ Weisstein, Eric W. "Surface Area". Wolfram MathWorld. Archived from the original on 23 June 2012. Retrieved 3 July 2012.
  7. ^ "Surface Area". CK-12 Foundation. Retrieved 2018-10-09.
  8. ^ do Carmo, Manfredo (1976). Differential Geometry of Curves and Surfaces. Prentice-Hall. p. 98, ISBN 978-0-13-212589-5
  9. ^ Walter Rudin (1966). Real and Complex Analysis, McGraw-Hill, ISBN 0-07-100276-6.
  10. ^ Gerald Folland (1999). Real Analysis: modern techniques and their applications, John Wiley & Sons, Inc., p. 20, ISBN 0-471-31716-0