áreas

Áreas is the plural of área in Spanish and Portuguese, used to refer to regions, zones, or extents of space. In mathematics, area denotes the two‑dimensional size of a region on the plane, expressed in square units such as square meters or square kilometers. The concept is grounded in basic properties: non-negativity, additivity over disjoint regions, and monotonicity with respect to inclusion. In more formal terms, area is tied to measure theory, where the area of simple shapes aligns with the Lebesgue measure in the plane, and the idea extends to higher dimensions as volume.

Common shapes have standard formulas for their areas. Rectangle or parallelogram: A = base × height. Triangle: A = 1/2 × base × height. Circle: A = πr^2, where r is the radius. Trapezoid: A = (base1 + base2)/2 × height. More complex figures can be analyzed by decomposing them into simpler parts or by integration. In calculus, area under a curve y = f(x) from x = a to x = b is A = ∫_a^b f(x) dx; for irregular regions, double integrals or other techniques from multivariable calculus are used.

Beyond mathematics, áre as is used in geography and planning to denote bounded portions of space, such as urban areas, rural areas, protected areas, or administrative districts. These areas are defined by legal boundaries or natural features and are used for mapping, statistics, environmental management, and governance. In education and research, áre as may also refer to domains of study or departmental divisions within institutions.

Overall, the term encompasses both quantitative measurements of space and qualitative designations of regions across disciplines.