areas

Area is a measure of the size of a two-dimensional surface or region. It quantifies the extent of a figure in the plane and is expressed in square units such as square meters, square kilometers, or square miles. Area is a fundamental concept in mathematics, geography, and many applied fields.

In geometry, area can be computed using simple formulas for common shapes: a rectangle has area equal to length times width, a triangle has area half the product of base and height, and a circle has area equal to pi times the radius squared. For irregular or complex shapes, area can be found by decomposing the shape into simpler parts, applying known formulas, or by using integral methods and the shoelace formula for polygons. Area is additive: the area of a region composed of non-overlapping parts is the sum of the areas of its parts.

Units and scaling are important aspects of area. Because area uses two dimensions, it scales with the square of linear measurements: if all linear dimensions are doubled, the area increases by a factor of four. Area is also sensitive to unit choice through conversions between square units (for example, 1 square meter equals 10,000 square centimeters).

Applications of area include mapping and land management, urban planning, ecology, and climate science, where the size of regions, parcels, or habitats is relevant. In geography and GIS, area helps compare regions, estimate resources, and model spatial phenomena. Area is distinct from volume, which measures three-dimensional space, and from surface area, which describes the boundary of a three-dimensional object.