area

Area is a measure of the extent of a two-dimensional surface or region. In mathematics, it denotes the size of a region in the plane and is expressed in square units, such as square meters (m^2) or square kilometers (km^2).

In Euclidean geometry, area can be computed by formulae for standard shapes: a rectangle has area A = length × width; a square is s^2; a triangle is (base × height)/2; a circle is A = πr^2; a trapezoid is ((b1 + b2)/2) × h. For polygons, area can be found by subdividing into triangles or using the shoelace formula. For irregular shapes, area can be measured by integration or by planimetry; in calculus, the area of a region R is the double integral over R of 1 dA.

Measurement methods vary. Practical approaches include counting grid squares or using planimeters. In calculus and analysis, area is related to integrals and can be defined as the measure of a set in the plane; more generally, the area of a parametric region can be computed via a surface integral.

Area is a two-dimensional measure, distinct from volume (the three-dimensional amount of space a solid occupies) and from surface area (the measure of a boundary surface). In higher mathematics, area corresponds to the two-dimensional Lebesgue measure; for parametrized surfaces, the area is obtained through a surface integral.

The concept has origins in ancient geometry and is central to disciplines such as land surveying, architecture, engineering, and physics. Area enables comparisons of size, resource estimation, and spatial analysis across diverse fields.