Area
Area is the measure of a region's size on a surface, and it sits at the heart of a problem that pushed mathematics forward for more than two thousand years. Think of it as the amount of paint needed to cover a surface with a single coat, or the amount of material with a given thickness needed to fashion a model of a shape. It is the two-dimensional companion to two other ideas you already sense intuitively: the length of a curve, which is one-dimensional, and the volume of a solid, which is three-dimensional.
The trouble starts the moment a boundary curves. A rectangle is easy. A circle, an ellipse, or the surface of a sphere refuses to surrender its size to simple multiplication. How did anyone first prove that the area inside a circle relates to the square of its diameter? Why did the chase for the area of plane figures end up inventing one of mathematics' most powerful tools? And how does a single concept stretch from the size of a farmer's field to the cross-section of a nuclear interaction measured in something called a barn? Those questions guide what follows.
Comparing a shape to squares of a fixed size is how area gets measured at its most basic. In the International System of Units, the standard unit is the square metre, written m2, defined as the area of a square whose sides are one metre long. A shape with an area of three square metres covers the same ground as three such squares laid out.
The unit square anchors the whole system in mathematics, defined to have area one, with the area of any other shape becoming a dimensionless real number. Every unit of length carries a matching unit of area, the square of a side with that length, which is why square centimetres, square kilometres, square feet, square yards, and square miles all exist.
Non-metric conversions follow a tidy rule: the conversion between two square units is the square of the conversion between the matching length units. Since 1 foot equals 12 inches, 1 square foot equals 144 square inches, because 144 is 12 times 12. One square yard works out to 9 square feet, and a single square mile swells to 27,878,400 square feet. That squaring rule means small differences in length explode into large differences in area, a fact that becomes vivid the moment land enters the picture.
The are was the original unit of area in the metric system, set at 100 square metres, and though it has fallen out of use, its larger cousin survives. One hectare equals 100 ares, or 10,000 square metres, and it remains the common way to measure land. Older metric oddities like the tetrad, the hectad, and the myriad linger on the margins.
The acre tells a less orderly story. One acre equals 4,840 square yards, or 43,560 square feet, and it works out to roughly 40% of a hectare. In South Asia, where countries use SI units officially, many people still reach for traditional measures, and the picture grows tangled. Each administrative division has its own area unit, some sharing names but carrying different values, with no official consensus on what those values are.
A few South Asian units do hold a fixed value. One killa equals one acre, as does one ghumaon, while one kanal is 0.125 acre, so eight kanal make an acre. A decimal comes to 48.4 square yards, and a chatak to 180 square feet. Far from the field, at the atomic scale, area shrinks to the barn, where one barn equals ten to the minus 28 square metres, used to describe the cross-sectional area of interaction in nuclear physics.
In the 5th century BCE, Hippocrates of Chios became the first to show that the area of a disk is proportional to the square of its diameter, work tied to his quadrature of the lune of Hippocrates. He stopped short of naming the constant of proportionality. In the same century, Eudoxus of Cnidus found that a disk's area is proportional to its radius squared.
Book I of Euclid's Elements then took up equality of areas between two-dimensional figures. Archimedes pushed further in his book Measurement of a Circle, using Euclidean geometry to show that the area inside a circle equals that of a right triangle whose base has the length of the circle's circumference and whose height equals the radius. Since the circumference is 2 pi r and a triangle's area is half the base times the height, this yields pi r squared for the disk.
Archimedes also pinned down the value of pi by a doubling method. He inscribed a regular triangle in a circle, noted its area, then doubled the sides to make a hexagon, and kept doubling as the polygon crept closer to the circle, repeating the trick with circumscribed polygons too. In ancient times this kind of reasoning was called the method of exhaustion, now recognized as a precursor to integral calculus.
In the 7th century CE, Brahmagupta developed a formula, now bearing his name, for the area of a cyclic quadrilateral, one inscribed in a circle, expressed through its sides. The reach of these quadrilateral results widened in 1842, when the German mathematicians Carl Anton Bretschneider and Karl Georg Christian von Staudt independently found a formula, known as Bretschneider's formula, for the area of any quadrilateral.
René Descartes changed the terrain in the 17th century by developing Cartesian coordinates. That advance opened the way for the surveyor's formula, which gives the area of any polygon from the locations of its vertices, worked out by Gauss in the 19th century. For a simple polygon whose n vertices are known, this formula reads off the area directly from the coordinates.
Knowing the vertices unlocks other tools too. The same coordinate approach is captured in the shoelace formula, which finds the area of a coordinate triangle by substituting its three points, and extends to other polygons whose vertices are known. Pick's theorem offers a different route entirely, computing the area of a polygon on a grid of equally spaced integer points from the count of grid points inside it and on its boundary.
The most basic area formula is the one for a rectangle: length multiplied by width. A square is the special case where the side length s gives the area directly. This formula follows from the basic properties of area and is sometimes taken as a definition or axiom; flip the logic, and if geometry comes before arithmetic, it can even define the multiplication of real numbers.
Most other simple formulas come from the method of dissection, cutting a shape into pieces whose areas must sum to the original. A parallelogram splits into a trapezoid and a right triangle, and sliding that triangle to the other side leaves a rectangle, so the parallelogram and rectangle share the same area. Cut the same parallelogram along a diagonal instead, and you get two congruent triangles, so each triangle holds half the parallelogram's area.
The circle yields to a cleverer cut. Partition it into sectors, each roughly triangular, and rearrange them into a shape close to a parallelogram whose width is half the circumference. The dissection is only approximate, but the error shrinks as the sectors multiply, and the limit lands exactly on the circle's area, a quiet application of the ideas of calculus. The same flattening trick handles non-planar surfaces too: slice a cylinder lengthwise and it opens into a rectangle, slice a cone and it opens into a sector of a circle.
A sphere cannot be flattened out, and that single fact makes its surface area genuinely hard to derive. The obstacle is its nonzero Gaussian curvature, which resists the cut-and-flatten method that tames a cylinder or a cone. Archimedes solved it anyway in his work On the Sphere and Cylinder, producing the first formula for a sphere's surface area, with the radius as its key quantity.
The broader pattern is that the area of a solid's boundary surface earns its own name, surface area, covering shapes like the sphere, cone, and cylinder. The ancient Greeks computed surface areas for simple shapes, but a more complicated shape usually demands multivariable calculus. There is a general formula for the surface area of the graph of a continuously differentiable function over a region in the xy-plane with a smooth boundary, and a still more general one for a parametric surface written as a continuously differentiable vector function.
Calculus broadened the whole enterprise once integral calculus arrived in the late 17th century. Suddenly the area of an ellipse and the surface areas of various curved three-dimensional objects came within reach. The ellipse's area connects directly to the circle's, written through its semi-major and semi-minor axes, a relationship that flows from the same definite-integral reasoning behind the circle.
Area is related to the definition of determinants in linear algebra, a connection that lifts it well beyond geometry class. It is a basic property of surfaces in differential geometry, and in analysis the area of a subset of the plane is defined using Lebesgue measure. Strikingly, not every subset is measurable if one supposes the axiom of choice, so some regions slip outside the reach of area altogether. In higher mathematics, area is seen as a special case of volume for two-dimensional regions.
The concept can be built from the ground up through axioms, defining area as a function from a collection of measurable plane figures to the real numbers. The rules demand that area never go negative, that congruent figures share an area, and that every rectangle of length h and breadth k be measurable. A clinching condition handles curved regions: a set squeezed between two step regions, each a finite union of adjacent rectangles, takes the unique value pinned between them. It can be proved that such an area function actually exists.
Area also speaks to shape itself. The isoperimetric inequality states that, for a closed curve of a given length, the enclosed area is largest exactly when the curve is a circle. Push the other way and a figure of fixed perimeter can have arbitrarily small area, like a rhombus tipped over until two angles near 0 degrees and two near 180 degrees. The same drive toward efficiency reappears in the open question of the filling area of the Riemannian circle, a minimal-surface puzzle that, like soap bubbles spanning a wire contour, remains unsolved.
Common questions
What is area in mathematics?
Area is the measure of a region's size on a surface. It is the two-dimensional analogue of the length of a curve, which is one-dimensional, and the volume of a solid, which is three-dimensional.
What is the SI unit of area?
The standard unit of area in the International System of Units is the square metre, written m2, which is the area of a square whose sides are one metre long. The square metre is considered an SI derived unit.
Who first proved the area of a circle relates to its diameter?
In the 5th century BCE, Hippocrates of Chios was the first to show that the area of a disk is proportional to the square of its diameter, though he did not identify the constant of proportionality. In the same century, Eudoxus of Cnidus found that a disk's area is proportional to its radius squared.
How did Archimedes find the area of a circle?
Archimedes showed in Measurement of a Circle that the area inside a circle equals that of a right triangle whose base has the length of the circle's circumference and whose height equals the radius, yielding pi r squared. He approximated pi by inscribing and circumscribing polygons and repeatedly doubling their number of sides.
How many square feet are in an acre?
One acre equals 4,840 square yards, or 43,560 square feet. An acre is approximately 40% of a hectare.
What is a barn unit of area used for?
A barn is a unit of area equal to ten to the minus 28 square metres, used on the atomic scale. The barn is commonly used in describing the cross-sectional area of interaction in nuclear physics.
Why is the surface area of a sphere hard to calculate?
A sphere has nonzero Gaussian curvature, so it cannot be flattened out the way a cylinder or cone can. The formula for the surface area of a sphere was first obtained by Archimedes in his work On the Sphere and Cylinder.