Area measures how much room a region covers on a surface
Plane area sizes a flat shape; surface area sizes the boundary of a solid. Intuitively it is how much thin material or one coat of paint would cover the figure—the two-dimensional cousin of length. Comparing regions to unit squares makes the idea precise, with the square metre as the SI standard.
Distinct regions can share one numerical size—the classic circle-squaring puzzle—and speakers sometimes use the word loosely for the region itself. Formulas for triangles, rectangles, and circles let one triangulate any polygon; curved boundaries usually need calculus, and that historical demand helped create the subject. Surface areas of spheres, cones, and cylinders were known to the Greeks, while complicated surfaces need multivariable calculus. Modern mathematics ties area to determinants, differential geometry of surfaces, and Lebesgue measure in the plane—though under the axiom of choice not every subset is measurable. Higher-dimensional volume generalizes the same idea.
Axiomatically, area is a real-valued function on measurable plane figures that is additive for unions after subtracting intersections, subtractive for differences when one set sits inside another, invariant under congruence, assigns length times breadth to rectangles, and squeezes sets trapped between step regions of rectangles to a unique value when such a number exists. Existence of such a function can be proved. Every length unit has a matching square unit: square metres, centimetres, kilometres, feet, and so on, algebraically the squares of the length units. A three-by-two-metre rectangle has area six square metres.
Conversions follow: one square kilometre is a million square metres; one square metre is ten thousand square centimetres. Non-metric square conversions square the linear factors—one square mile equals 27,878,400 square feet. The are was the original metric land unit; the hectare equals 100 ares or 10,000 square metres and remains common. An acre is 43,560 square feet, roughly forty percent of a hectare. Nuclear physics measures tiny cross sections in barns. South Asian traditional land units vary by region even when SI is official, so conversions depend on local references.
Historically, Hippocrates of Chios showed in the fifth century BCE that a disk's area is proportional to the square of its diameter without naming the constant; Eudoxus found proportionality to the radius squared. Euclid's Elements Book I treated equality of areas. Archimedes proved the disk's area equals that of a right triangle whose base is the circumference and whose height is the radius, recovering π r squared. Area thus sits between everyday covering and deep measure theory—the same quantity that paints a wall also defines what countable additivity must respect.
Source: Area
