Area is themeasure of aregion's size on asurface. The area of a plane region orplane area refers to the area of ashape orplanar lamina, whilesurface area refers to the area of anopen surface or theboundary of athree-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount ofpaint necessary to cover the surface with a single coat.[1] It is the two-dimensional analogue of thelength of acurve (a one-dimensional concept) or thevolume of a solid (a three-dimensional concept).Two different regions may have the same area (as insquaring the circle); bysynecdoche, "area" sometimes is used to refer to the region, as in a "polygonal area".
The area of a shape can be measured by comparing the shape tosquares of a fixed size.[2] In theInternational System of Units (SI), the standard unit of area is thesquare metre (written as m2), which is the area of a square whose sides are onemetre long.[3] A shape with an area of three square metres would have the same area as three such squares. Inmathematics, theunit square is defined to have area one, and the area of any other shape or surface is adimensionlessreal number.
For a solid shape such as asphere, cone, or cylinder, the area of its boundary surface is called thesurface area.[1][6][7] Formulas for the surface areas of simple shapes were computed by theancient Greeks, but computing the surface area of a more complicated shape usually requiresmultivariable calculus.
Area plays an important role in modern mathematics. In addition to its obvious importance ingeometry and calculus, area is related to the definition ofdeterminants inlinear algebra, and is a basic property of surfaces indifferential geometry.[8] Inanalysis, the area of a subset of the plane is defined usingLebesgue measure,[9] though not every subset is measurable if one supposes the axiom of choice.[10] In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.[1]
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 throughaxioms. "Area" can be defined as a function from a collection M of a special kinds of plane figures (termed measurable sets) to the set of real numbers, which satisfies the following properties:[11]
For allS inM,a(S) ≥ 0.
IfS andT are inM then so areS ∪T andS ∩T, and alsoa(S∪T) =a(S) +a(T) −a(S ∩T).
IfS andT are inM withS ⊆T thenT −S is inM anda(T−S) =a(T) −a(S).
If a setS is inM andS is congruent toT thenT is also inM anda(S) =a(T).
Every rectangleR is inM. If the rectangle has lengthh and breadthk thena(R) =hk.
LetQ be a set enclosed between two step regionsS andT. A step region is formed from a finite union of adjacent rectangles resting on a common base, i.e.S ⊆Q ⊆T. If there is a unique numberc such thata(S) ≤ c ≤a(T) for all such step regionsS andT, thena(Q) =c.
It can be proved that such an area function actually exists.[12]
Everyunit of length has a corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measured insquare metres (m2), square centimetres (cm2), square millimetres (mm2),square kilometres (km2),square feet (ft2),square yards (yd2),square miles (mi2), and so forth.[13] Algebraically, these units can be thought of as thesquares of the corresponding length units.
The SI unit of area is the square metre, which is considered anSI derived unit.[3]
Other uncommon metric units of area include thetetrad, thehectad, and themyriad.
Theacre is also commonly used to measure land areas, where
1 acre = 4,840 square yards = 43,560 square feet.
An acre is approximately 40% of a hectare.
On the atomic scale, area is measured in units ofbarns, such that:[13]
1 barn = 10−28 square meters.
The barn is commonly used in describing the cross-sectional area of interaction innuclear physics.[13]
InSouth Asia (mainly Indians), although the countries use SI units as official, many South Asians still use traditional units. Each administrative division has its own area unit, some of them have same names, but with different values. There's no official consensus about the traditional units values. Thus, the conversions between the SI units and the traditional units may have different results, depending on what reference that has been used.[14][15][16][17]
Some traditional South Asian units that have fixed value:
In the 5th century BCE,Hippocrates of Chios was the first to show that the area of a disk (the region enclosed by a circle) is proportional to the square of its diameter, as part of hisquadrature of thelune of Hippocrates,[18] but did not identify theconstant of proportionality.Eudoxus of Cnidus, also in the 5th century BCE, also found that the area of a disk is proportional to its radius squared.[19]
Subsequently, Book I ofEuclid'sElements dealt with equality of areas between two-dimensional figures. The mathematicianArchimedes used the tools ofEuclidean geometry to show that the area inside a circle is equal to that of aright triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, in his bookMeasurement of a Circle. (The circumference is 2πr, and the area of a triangle is half the base times the height, yielding the areaπr2 for the disk.) Archimedes approximated the value ofπ (and hence the area of a unit-radius circle) withhis doubling method, in which he inscribed a regular triangle in a circle and noted its area, then doubled the number of sides to give a regularhexagon, then repeatedly doubled the number of sides as the polygon's area got closer and closer to that of the circle (and did the same withcircumscribed polygons).
Heron of Alexandria found what is known asHeron's formula for the area of a triangle in terms of its sides, and a proof can be found in his book,Metrica, written around 60 CE. It has been suggested thatArchimedes knew the formula over two centuries earlier,[20] and sinceMetrica is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work.[21] In 300 BCE Greek mathematicianEuclid proved that the area of a triangle is half that of a parallelogram with the same base and height in his bookElements of Geometry.[22]
The development ofintegral calculus in the late 17th century provided tools that could subsequently be used for computing more complicated areas, such as the area of anellipse and thesurface areas of various curved three-dimensional objects.
The most basic area formula is the formula for the area of arectangle. Given a rectangle with lengthl and widthw, the formula for the area is:[2]
A =lw (rectangle).
That is, the area of the rectangle is the length multiplied by the width. As a special case, asl =w in the case of a square, the area of a square with side lengths is given by the formula:[1][2]
A =s2 (square).
The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as adefinition oraxiom. On the other hand, ifgeometry is developed beforearithmetic, this formula can be used to definemultiplication ofreal numbers.
A parallelogram can be cut up and re-arranged to form a rectangle.
Most other simple formulas for area follow from the method ofdissection.This involves cutting a shape into pieces, whose areas mustsum to the area of the original shape.For an example, anyparallelogram can be subdivided into atrapezoid and aright triangle, as shown in figure to the left. If the triangle is moved to the other side of the trapezoid, then the resulting figure is a rectangle. It follows that the area of the parallelogram is the same as the area of the rectangle:[2]
A =bh (parallelogram).
A parallelogram split into two equal triangles
However, the same parallelogram can also be cut along adiagonal into twocongruent triangles, as shown in the figure to the right. It follows that the area of eachtriangle is half the area of the parallelogram:[2]
(triangle).
Similar arguments can be used to find area formulas for thetrapezoid[26] as well as more complicatedpolygons.[27]
The formula for the area of acircle (more properly called the area enclosed by a circle or the area of adisk) is based on a similar method. Given a circle of radiusr, it is possible to partition the circle intosectors, as shown in the figure to the right. Each sector is approximately triangular in shape, and the sectors can be rearranged to form an approximate parallelogram. The height of this parallelogram isr, and the width is half thecircumference of the circle, orπr. Thus, the total area of the circle isπr2:[2]
A = πr2 (circle).
Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors. Thelimit of the areas of the approximate parallelograms is exactlyπr2, which is the area of the circle.[28]
This argument is actually a simple application of the ideas ofcalculus. In ancient times, themethod of exhaustion was used in a similar way to find the area of the circle, and this method is now recognized as a precursor tointegral calculus. Using modern methods, the area of a circle can be computed using adefinite integral:
The formula for the area enclosed by anellipse is related to the formula of a circle; for an ellipse withsemi-major andsemi-minor axesx andy the formula is:[2]
Archimedes showed that the surface area of asphere is exactly four times the area of a flatdisk of the same radius, and the volume enclosed by the sphere is exactly 2/3 of the volume of acylinder of the same height and radius.
Most basic formulas forsurface area can be obtained by cutting surfaces and flattening them out (see:developable surfaces). For example, if the side surface of acylinder (or anyprism) is cut lengthwise, the surface can be flattened out into a rectangle. Similarly, if a cut is made along the side of acone, the side surface can be flattened out into a sector of a circle, and the resulting area computed.
The formula for the surface area of asphere is more difficult to derive: because a sphere has nonzeroGaussian curvature, it cannot be flattened out. The formula for the surface area of a sphere was first obtained byArchimedes in his workOn the Sphere and Cylinder. The formula is:[6]
A = 4πr2 (sphere),
wherer 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 tocalculus.
Atriangle: (whereB is any side, andh is the distance from the line on whichB lies to the other vertex of the triangle). This formula can be used if the heighth is known. If the lengths of the three sides are known thenHeron's formula can be used: wherea,b,c are the sides of the triangle, and is half of its perimeter.[2] If an angle and its two included sides are given, the area is whereC is the given angle anda andb are its included sides.[2] If the triangle is graphed on a coordinate plane, a matrix can be used and is simplified to the absolute value of. This formula is also known as theshoelace formula and is an easy way to solve for the area of a coordinate triangle by substituting the 3 points(x1,y1),(x2,y2), and(x3,y3). The shoelace formula can also be used to find the areas of other polygons when their vertices are known. Another approach for a coordinate triangle is to usecalculus to find the area.
Asimple polygon constructed on a grid of equal-distanced points (i.e., points withinteger coordinates) such that all the polygon's vertices are grid points:, wherei is the number of grid points inside the polygon andb is the number of boundary points. This result is known asPick's theorem.[29]
Integration can be thought of as measuring the area under a curve, defined byf(x), between two points (herea andb).The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions
The area between a positive-valued curve and the horizontal axis, measured between two valuesa andb (b is defined as the larger of the two values) on the horizontal axis, is given by the integral froma tob of the function that represents the curve:[1]
The area between thegraphs of two functions isequal to theintegral of onefunction,f(x),minus the integral of the other function,g(x):
To find the bounded area between twoquadratic functions, we first subtract one from the other, writing the difference aswheref(x) is the quadratic upper bound andg(x) is the quadratic lower bound. By the area integral formulas above andVieta's formula, we can obtain that[30][31]The above remains valid if one of the bounding functions is linear instead of quadratic.
Cone:[32], wherer is the radius of the circular base, andh is the height. That can also be rewritten as[32] or wherer is the radius andl is the slant height of the cone. is the base area while is the lateral surface area of the cone.[32]
The general formula for the surface area of the graph of a continuously differentiable function where and is a region in the xy-plane with the smooth boundary:
An even more general formula for the area of the graph of aparametric surface in the vector form where is a continuously differentiable vector function of is:[8]
Theisoperimetric inequality states that, for a closed curve of lengthL (so the region it encloses hasperimeterL) and for areaA of the region that it encloses,
and equality holds if and only if the curve is acircle. Thus a circle has the largest area of any closed figure with a given perimeter.
At the other extreme, a figure with given perimeterL could have an arbitrarily small area, as illustrated by arhombus that is "tipped over" arbitrarily far so that two of itsangles are arbitrarily close to 0° and the other two are arbitrarily close to 180°.
For a circle, the ratio of the area to thecircumference (the term for the perimeter of a circle) equals half theradiusr. This can be seen from the area formulaπr2 and the circumference formula 2πr.
The area of aregular polygon is half its perimeter times theapothem (where the apothem is the distance from the center to the nearest point on any side).
Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). But if the one-dimensional lengths of afractal drawn in two dimensions are all doubled, the spatial content of the fractal scales by a power of two that is not necessarily an integer. This power is called thefractal dimension of the fractal.[33]
There are an infinitude of lines that bisect the area of a triangle. Three of them are themedians of the triangle (which connect the sides' midpoints with the opposite vertices), and these areconcurrent at the triangle'scentroid; indeed, they are the only area bisectors that go through the centroid. Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of itsincircle). There are either one, two, or three of these for any given triangle.
Any line through the midpoint of a parallelogram bisects the area.
All area bisectors of a circle or other ellipse go through the center, and anychords through the center bisect the area. In the case of a circle they are the diameters of the circle.
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.[36]
The ratio of the area of the incircle to the area of an equilateral triangle,, is larger than that of any non-equilateral triangle.[37]
The ratio of the area to the square of the perimeter of an equilateral triangle, is larger than that for any other triangle.[35]
^abChakerian, G.D. (1979) "A Distorted View of Geometry." Ch. 7 inMathematical Plums. R. Honsberger (ed.). Washington, DC: Mathematical Association of America, p. 147.
^Dorrie, Heinrich (1965),100 Great Problems of Elementary Mathematics, Dover Publ., pp. 379–380.
