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Area: the story on HearLore | HearLore
Area
Area is not merely a number on a page but the fundamental measure of how much space a shape occupies on a flat surface. This concept serves as the two-dimensional analogue to the length of a curve and the volume of a solid, acting as the bridge between simple geometry and complex mathematical analysis. In the International System of Units, the standard unit is the square metre, defined as the area of a square with sides of one metre, yet the concept extends far beyond the metric system to include square feet, square miles, and even the barn, a unit used in nuclear physics to describe the cross-sectional area of interaction. The area of a shape can be visualized as the amount of paint necessary to cover a surface with a single coat or the amount of material with a given thickness required to fashion a model of that shape. This definition allows mathematicians to treat area as a dimensionless real number when using the unit square, where the area of any other shape is simply a comparison to that standard. The problem of determining the area of plane figures was a major motivation for the historical development of calculus, driving centuries of inquiry into how to measure the curved and the irregular.
Ancient Geometers and the Circle
In the 5th century BCE, Hippocrates of Chios became the first to demonstrate that the area of a disk is proportional to the square of its diameter, a breakthrough achieved during his work on the quadrature of the lune of Hippocrates. Although he did not identify the constant of proportionality, his work laid the groundwork for future discoveries. Eudoxus of Cnidus, also in the 5th century BCE, refined this understanding by proving that the area of a disk is proportional to its radius squared. The mathematician Archimedes took these concepts further in his book Measurement of a Circle, using the tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius. Archimedes approximated the value of pi with his doubling method, inscribing a regular triangle in a circle and repeatedly doubling the number of sides to make the polygon's area get closer and closer to that of the circle. This method of exhaustion, used in ancient times to find the area of the circle, is now recognized as a precursor to integral calculus. The ancient Greeks computed formulas for the surface areas of simple shapes, but the surface area of a more complicated shape usually requires multivariable calculus to compute.
The Evolution of Formulas
The development of area formulas has evolved from simple dissection methods to complex algebraic expressions. In the 7th century CE, Brahmagupta developed a formula for the area of a cyclic quadrilateral, a quadrilateral inscribed in a circle, in terms of its sides. In 1842, the German mathematicians Carl Anton Bretschneider and Karl Georg Christian von Staudt independently found a formula for the area of any quadrilateral. The development of Cartesian coordinates by René Descartes in the 17th century allowed the development of the surveyor's formula for the area of any polygon with known vertex locations by Gauss in the 19th century. Most other simple formulas for area follow from the method of dissection, which involves cutting a shape into pieces whose areas must sum to the area of the original shape. For example, any parallelogram can be subdivided into a trapezoid and a right triangle, and if the triangle is moved to the other side of the trapezoid, the resulting figure is a rectangle. This logic allows for the derivation of area formulas for the trapezoid as well as more complicated polygons. The formula for the area of a circle is based on a similar method, where the circle is partitioned into sectors that can be rearranged to form an approximate parallelogram, with the error becoming smaller and smaller as the circle is partitioned into more and more sectors.
Common questions
What is the standard unit of area in the International System of Units?
The standard unit of area in the International System of Units is the square metre. This unit is defined as the area of a square with sides of one metre and serves as the SI derived unit for area.
Who first demonstrated that the area of a disk is proportional to the square of its diameter?
Hippocrates of Chios became the first to demonstrate that the area of a disk is proportional to the square of its diameter in the 5th century BCE. His work on the quadrature of the lune of Hippocrates laid the groundwork for future discoveries in geometry.
What is the formula for the surface area of a sphere?
The formula for the surface area of a sphere is 4 pi r squared, where r is the radius of the sphere. Archimedes first obtained this formula in his work On the Sphere and Cylinder, showing that the surface area of a sphere is exactly four times the area of a flat disk of the same radius.
How is the area of a cyclic quadrilateral calculated?
Brahmagupta developed a formula for the area of a cyclic quadrilateral in the 7th century CE in terms of its sides. A cyclic quadrilateral is defined as a quadrilateral inscribed in a circle.
What does the isoperimetric inequality state about a circle?
The isoperimetric inequality states that a circle has the largest area of any closed figure with a given perimeter. For a closed curve of length L and area A, 4 pi A is less than or equal to L squared, and equality holds if and only if the curve is a circle.
Every unit of length has a corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measured in square metres, square centimetres, square millimetres, square kilometres, square feet, square yards, square miles, and so forth. Algebraically, these units can be thought of as the squares of the corresponding length units. The SI unit of area is the square metre, which is considered an SI derived unit. In non-metric units, the conversion between two square units is the square of the conversion between the corresponding length units. For instance, since 1 foot equals 12 inches, the relationship between square feet and square inches is 1 square foot equals 144 square inches, where 144 equals 12 squared. Other useful conversions include 1 square kilometre equals 1,000,000 square metres and 1 square metre equals 10,000 square centimetres. The are was the original unit of area in the metric system, with 1 are equal to 100 square metres. Though the are has fallen out of use, the hectare is still commonly used to measure land, where 1 hectare equals 100 ares or 10,000 square metres. The acre is also commonly used to measure land areas, where 1 acre equals 4,840 square yards or 43,560 square feet, which is approximately 40% of a hectare. On the atomic scale, area is measured in units of barns, such that 1 barn equals 10 to the power of negative 28 square meters, commonly used in describing the cross-sectional area of interaction in nuclear physics.
Curved Surfaces and Calculus
The formula for the surface area of a sphere is more difficult to derive because a sphere has nonzero Gaussian curvature and cannot be flattened out. The formula for the surface area of a sphere was first obtained by Archimedes in his work On the Sphere and Cylinder, showing that the surface area of a sphere is exactly four times the area of a flat disk of the same radius. The formula is 4 pi r squared, where r is the radius of the sphere. As with the formula for the area of a circle, any derivation of this formula inherently uses methods similar to calculus. The development of integral calculus in the late 17th century provided tools that could subsequently be used for computing more complicated areas, such as the area of an ellipse and the surface areas of various curved three-dimensional objects. The area between a positive-valued curve and the horizontal axis, measured between two values a and b on the horizontal axis, is given by the integral from a to b of the function that represents the curve. The area between the graphs of two functions is equal to the integral of one function minus the integral of the other function. An area bounded by a function expressed in polar coordinates is given by a specific integral, and the area enclosed by a parametric curve is given by line integrals. This is the principle of the planimeter mechanical device, an instrument for measuring small areas, such as on maps.
Optimization and Inequalities
The isoperimetric inequality states that for a closed curve of length L and for area A of the region that it encloses, 4 pi A is less than or equal to L squared, and equality holds if and only if the curve is a circle. Thus a circle has the largest area of any closed figure with a given perimeter. At the other extreme, a figure with given perimeter L could have an arbitrarily small area, as illustrated by a rhombus that is tipped over arbitrarily far so that two of its angles are arbitrarily close to 0 degrees and the other two are arbitrarily close to 180 degrees. Given a wire contour, the surface of least area spanning it is a minimal surface, with familiar examples including soap bubbles. The circle has the largest area of any two-dimensional object having the same perimeter. A cyclic polygon, one inscribed in a circle, has the largest area of any polygon with a given number of sides of the same lengths. A version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral. The triangle of largest area of all those inscribed in a given circle is equilateral, and the triangle of smallest area of all those circumscribed around a given circle is equilateral. The ratio of the area of the incircle to the area of an equilateral triangle is larger than that of any non-equilateral triangle.
Modern Mathematics and Axioms
Area plays an important role in modern mathematics, related to the definition of determinants in linear algebra and serving as a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, though not every subset is measurable if one supposes the axiom of choice. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists. An approach to defining what is meant by area is through axioms, where area is defined as a function from a collection M of a special kinds of plane figures to the set of real numbers, which satisfies specific properties. For all S in M, the area is non-negative. If S and T are in M, then so are their union and intersection, and also their difference. If a set S is in M and S is congruent to T, then T is also in M and has the same area. Every rectangle R is in M, and if the rectangle has length h and breadth k, then its area is hk. Let Q be a set enclosed between two step regions S and T, where a step region is formed from a finite union of adjacent rectangles resting on a common base. If there is a unique number c such that the area of S is less than or equal to c and c is less than or equal to the area of T for all such step regions S and T, then the area of Q is c. It can be proved that such an area function actually exists.