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In
geometry Geometry (from the grc, γεωμετρία; ''geo-'' "earth", ''-metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, and ...
, the
area Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of ...

area
enclosed by a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. T ...
of
radius In classical geometry, a radius of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the Latin ''radius'', meaning ray but also the spoke of a c ...
is . Here the Greek letter represents the
constant Constant or The Constant may refer to: Mathematics * Constant (mathematics), a non-varying value * Mathematical constant, a special number that arises naturally in mathematics, such as or Other concepts * Control variable, in experimentation, ...
ratio of the
circumference In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out to a l ...
of any circle to its
diameter In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for th ...
, approximately equal to 3.1416. One method of deriving this formula, which originated with
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Considered to be ...

Archimedes
, involves viewing the circle as the limit of a sequence of
regular polygon In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex or star. In the limit, a sequence of regular polygon ...
s. The area of a regular polygon is half its
perimeter A perimeter is either a path that encompasses/surrounds/outlines a shape (in two dimensions) or its length (one-dimensional). The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several practical ...
multiplied by the distance from its center to its sides, and the corresponding formula–that the area is half the perimeter times the radius–namely, , holds in the limit for a circle. Although often referred to as the area of a circle in informal contexts, strictly speaking the term ''
disk Disk or disc may refer to: * Disk (mathematics) * Disk storage Music * Disc (band), an American experimental music band * ''Disk'' (album), a 1995 EP by Moby Other uses * Disc (galaxy), a disc-shaped group of stars * ''Disc'' (magazine), a Briti ...
'' refers to the interior of the circle, while ''circle'' is reserved for the boundary only, which is a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that app ...
and covers no area itself. Therefore, the area of a disk is the more precise phrase for the area enclosed by a circle.


History

Modern mathematics can obtain the area using the methods of
integral calculus In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with diffe ...
or its more sophisticated offspring,
real analysis 200px, The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis. In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, ...
. However, the area of a disk was studied by the
Ancient Greeks Ancient Greece ( el, Ἑλλάς, Hellás) was a civilization belonging to a period of Greek history from the Greek Dark Ages of the 12th–9th centuries BC to the end of antiquity ( AD 600). This era was immediately followed by the Early Middle ...
.
Eudoxus of Cnidus Eudoxus of Cnidus (; grc, Εὔδοξος ὁ Κνίδιος, ''Eúdoxos ho Knídios''; ) was an ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato. All of his works are lost, though some fragments are preserved in ...
in the fifth century B.C. had found that the area of a disk is proportional to its radius squared.
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Considered to be ...

Archimedes
used the tools of
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the ''Elements''. Euclid's method consists in assuming a small set of intuitively appealing axioms, ...
to show that the area inside a circle is equal to that of a
right triangle A right triangle (American English) or right-angled triangle (British ) is a triangle in which one angle is a right angle (that is, a 90-degree angle). The relation between the sides and angles of a right triangle is the basis for trigonometry. T ...
whose base has the length of the circle's circumference and whose height equals the circle's radius in his book ''
Measurement of a Circle ''Measurement of a Circle'' or ''Dimension of the Circle'' (Greek: , ''Kuklou metrēsis'') is a treatise that consists of three propositions by Archimedes, ca. 250 BCE. The treatise is only a fraction of what was a longer work. Propositions Propos ...
''. The circumference is 2''r'', and the area of a triangle is half the base times the height, yielding the area  ''r''2 for the disk. Prior to Archimedes,
Hippocrates of Chios Hippocrates of Chios ( grc-gre, Ἱπποκράτης ὁ Χῖος; c. 470 – c. 410 BC) was an ancient Greek mathematician, geometer, and astronomer. He was born on the isle of Chios, where he was originally a merchant. After some misadventu ...
was the first to show that the area of a disk is proportional to the square of its diameter, as part of his quadrature of the
lune of Hippocrates In geometry, the lune of Hippocrates, named after Hippocrates of Chios, is a lune bounded by arcs of two circles, the smaller of which has as its diameter a chord spanning a right angle on the larger circle. Equivalently, it is a non-convex plane ...
,. but did not identify the
constant of proportionality In mathematics, two varying quantities are said to be in a relation of proportionality, multiplicatively connected to a constant; that is, when either their ratio or their product yields a constant. The value of this constant is called the coef ...
.


Historical arguments

A variety of arguments have been advanced historically to establish the equation A=\pi r^2 to varying degrees of mathematical rigor. The most famous of these is Archimedes'
method of exhaustion The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area betwe ...
, one of the earliest uses of the mathematical concept of a limit, as well as the origin of
Archimedes' axiom
Archimedes' axiom
which remains part of the standard analytical treatment of the
real number system Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...
. The original proof of Archimedes is not rigorous by modern standards, because it assumes that we can compare the length of arc of a circle to the length of a secant and a tangent line, and similar statements about the area, as geometrically evident.


Using polygons

The area of a
regular polygon In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex or star. In the limit, a sequence of regular polygon ...
is half its perimeter times the
apothem Apothem of a hexagon The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to ...
. As the number of sides of the regular polygon increases, the polygon tends to a circle, and the apothem tends to the radius. This suggests that the area of a disk is half the circumference of its bounding circle times the radius.


Archimedes's proof

Following Archimedes' argument in ''The Measurement of a Circle'' (c. 260 BCE), compare the area enclosed by a circle to a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius. If the area of the circle is not equal to that of the triangle, then it must be either greater or less. We eliminate each of these by contradiction, leaving equality as the only possibility. We use
regular polygon In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex or star. In the limit, a sequence of regular polygon ...
s in the same way.


Not greater

Suppose that the area ''C'' enclosed by the circle is greater than the area ''T'' = 12''cr'' of the triangle. Let ''E'' denote the excess amount.
Inscribe{{unreferenced, date=August 2012 frame, Inscribed circles of various polygons An inscribed triangle of a circle In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To s ...
a square in the circle, so that its four corners lie on the circle. Between the square and the circle are four segments. If the total area of those gaps, ''G''4, is greater than ''E'', split each arc in half. This makes the inscribed square into an inscribed octagon, and produces eight segments with a smaller total gap, ''G''8. Continue splitting until the total gap area, ''Gn'', is less than ''E''. Now the area of the inscribed polygon, ''Pn'' = ''C'' − ''Gn'', must be greater than that of the triangle. :\begin E &= C - T \\ &> G_n \\ P_n &= C - G_n \\ &> C - E \\ P_n &> T \end But this forces a contradiction, as follows. Draw a perpendicular from the center to the midpoint of a side of the polygon; its length, ''h'', is less than the circle radius. Also, let each side of the polygon have length ''s''; then the sum of the sides, ''ns'', is less than the circle circumference. The polygon area consists of ''n'' equal triangles with height ''h'' and base ''s'', thus equals 12''nhs''. But since ''h'' < ''r'' and ''ns'' < ''c'', the polygon area must be less than the triangle area, 12''cr'', a contradiction. Therefore, our supposition that ''C'' might be greater than ''T'' must be wrong.


Not less

Suppose that the area enclosed by the circle is less than the area ''T'' of the triangle. Let ''D'' denote the deficit amount. Circumscribe a square, so that the midpoint of each edge lies on the circle. If the total area gap between the square and the circle, ''G''4, is greater than ''D'', slice off the corners with circle tangents to make a circumscribed octagon, and continue slicing until the gap area is less than ''D''. The area of the polygon, ''Pn'', must be less than ''T''. :\begin D &= T - C \\ &> G_n \\ P_n &= C + G_n \\ &< C + D \\ P_n &< T \end This, too, forces a contradiction. For, a perpendicular to the midpoint of each polygon side is a radius, of length ''r''. And since the total side length is greater than the circumference, the polygon consists of ''n'' identical triangles with total area greater than ''T''. Again we have a contradiction, so our supposition that ''C'' might be less than ''T'' must be wrong as well. Therefore, it must be the case that the area enclosed by the circle is precisely the same as the area of the triangle. This concludes the proof.


Rearrangement proof

Following Satō Moshun and
Leonardo da Vinci Leonardo da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, sculptor, architect, draughtsman, theorist, engineer and scientist. While his fame initially rested on his achievements ...
, we can use inscribed regular polygons in a different way. Suppose we inscribe a
hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon or 6-gon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A ''regular hexagon' ...
. Cut the hexagon into six triangles by splitting it from the center. Two opposite triangles both touch two common diameters; slide them along one so the radial edges are adjacent. They now form a
parallelogram In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal ...
, with the hexagon sides making two opposite edges, one of which is the base, ''s''. Two radial edges form slanted sides, and the height, ''h'' is equal to its
apothem Apothem of a hexagon The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to ...
(as in the Archimedes proof). In fact, we can also assemble all the triangles into one big parallelogram by putting successive pairs next to each other. The same is true if we increase it to eight sides and so on. For a polygon with 2''n'' sides, the parallelogram will have a base of length ''ns'', and a height ''h''. As the number of sides increases, the length of the parallelogram base approaches half the circle circumference, and its height approaches the circle radius. In the limit, the parallelogram becomes a rectangle with width ''r'' and height ''r''. :


Modern proofs

There are various equivalent definitions of the constant π. The conventional definition in pre-calculus geometry is the ratio of the circumference of a circle to its diameter: :\pi=\frac. However, because the circumference of a circle is not a primitive analytical concept, this definition is not suitable in modern rigorous treatments. A standard modern definition is that is equal to twice the least positive root of the
cosine In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...
function or, equivalently, the half-period of the
sine In mathematics, the sine is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle, to the l ...

sine
(or cosine) function. The cosine function can be defined either as a
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c)^1 + a_2 (x - c)^2 + \cdots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
, or as the solution of a certain
differential equation In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the diffe ...
. This avoids any reference to circles in the definition of , so that statements about the relation of to the circumference and area of circles are actually theorems, rather than definitions, that follow from the analytical definitions of concepts like "area" and "circumference". The analytical definitions are seen to be equivalent, if it is agreed that the circumference of the circle is measured as a
rectifiable curve Arc length is the distance between two points along a section of a curve. Determining the length of an irregular arc segment is also called of a curve. The advent of infinitesimal calculus led to a general formula that provides closed-form solu ...
by means of the
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with diffe ...
:C = 2\int_^R \frac = 2R\int_^1\frac. The integral appearing on the right is an
abelian integralIn mathematics, an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form :\int_^z R(x,w) \, dx, where R(x,w) is an arbitrary rational function of the two variables x and w, whic ...
whose value is a half-period of the
sine In mathematics, the sine is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle, to the l ...

sine
function, equal to . Thus C=2\pi R=\pi D is seen to be true as a theorem. Several of the arguments that follow use only concepts from elementary calculus to reproduce the formula A=\pi r^2, but in many cases to regard these as actual proofs, they rely implicitly on the fact that one can develop trigonometric functions and the fundamental constant in a way that is totally independent of their relation to geometry. We have indicated where appropriate how each of these proofs can be made totally independent of all trigonometry, but in some cases that requires more sophisticated mathematical ideas than those afforded by elementary calculus.


Onion proof

Using calculus, we can sum the area incrementally, partitioning the disk into thin concentric rings like the layers of an
onion The onion (''Allium cepa'' L., from Latin ''cepa'' "onion"), also known as the bulb onion or common onion, is a vegetable that is the most widely cultivated species of the genus ''Allium''. The shallot is a botanical variety of the onion. Until ...
. This is the method of
shell integration Shell integration (the shell method in integral calculus) is a method for calculating the volume of a solid of revolution, when integrating along an axis ''perpendicular to'' the axis of revolution. This is in contrast to disc integration which i ...

shell integration
in two dimensions. For an infinitesimally thin ring of the "onion" of radius ''t'', the accumulated area is 2''t dt'', the circumferential length of the ring times its infinitesimal width (one can approximate this ring by a rectangle with width=2''t'' and height=''dt''). This gives an elementary integral for a disk of radius ''r''. :\begin \mathrm(r) &= \int_0^ 2 \pi t \, dt \\ &= 2\pi \left frac \right^\\ &= \pi r^2. \end It is rigorously justified by the multivariate substitution rule in polar coordinates. Namely, the area is given by a
double integral In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number pl ...
of the constant function 1 over the disk itself. If ''D'' denotes the disk, then the double integral can be computed in
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
as follows: :\begin \mathrm(r) &= \iint_D 1\ d(x, y)\\ & = \iint_D t\ dt\ d\theta\\ & = \int_0^r \int_0^ t\ d\theta\ dt\\ & = \int_0^r \left t\theta \right^ dt\\ &= \int_0^ 2 \pi t \, dt \\ \end which is the same result as obtained above. An equivalent rigorous justification, without relying on the special coordinates of trigonometry, uses the
coarea formulaIn the mathematical field of geometric measure theory, the coarea formula expresses the integral of a function over an open set in Euclidean space in terms of integrals over the level sets of another function. A special case is Fubini's theorem, whi ...
. Define a function \rho:\mathbb R^2\to\mathbb R by \rho(x,y)=\sqrt. Note ρ is a
Lipschitz function In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such ...
whose
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the vector whose components are the partial derivatives of f a ...

gradient
is a unit vector , \nabla\rho, =1 (
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to t ...
). Let ''D'' be the disc \rho<1 in \mathbb R^2. We will show that \mathcal L^2(D)=\pi, where \mathcal L^2 is the two-dimensional Lebesgue measure in \mathbb R^2. We shall assume that the one-dimensional
Hausdorff measure In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assig ...
of the circle \rho=r is 2\pi r, the circumference of the circle of radius ''r''. (This can be taken as the definition of circumference.) Then, by the coarea formula, :\begin\mathcal L^2(D) &= \iint_D , \nabla \rho, \,d\mathcal^2\\ &= \int_ \mathcal H^1(\rho^(r)\cap D)\,dr\\ &= \int_0^1\mathcal H^1(\rho^(r))\,dr \\ &= \int_0^1 2\pi r\, dr= \pi. \end


Triangle proof

Similar to the onion proof outlined above, we could exploit calculus in a different way in order to arrive at the formula for the area of a disk. Consider unwrapping the concentric circles to straight strips. This will form a right angled triangle with r as its height and 2r (being the outer slice of onion) as its base. Finding the area of this triangle will give the area of the disk :\begin \text &= \frac \cdot \text \cdot \text \\ &= \frac \cdot 2 \pi r \cdot r \\ &= \pi r^2 \end The opposite and adjacent angles for this triangle are respectively in degrees 9.0430611..., 80.956939... and in radians 0.1578311... , 1.4129651.... Explicitly, we imagine dividing up a circle into triangles, each with a height equal to the circle's radius and a base that is infinitesimally small. The area of each of these triangles is equal to 1/2\cdot r \cdot du. By summing up (integrating) all of the areas of these triangles, we arrive at the formula for the circle's area: :\begin \mathrm(r) &= \int_0^ \frac r \, du \\ &= \left \frac r u \right^\\ &= \pi r^2. \end It too can be justified by a double integral of the constant function 1 over the disk by reversing the
order of integration In statistics, the order of integration, denoted ''I''(''d''), of a time series is a summary statistic, which reports the minimum number of differences required to obtain a covariance-stationary series. Integration of order zero A time series is ...
and using a change of variables in the above iterated integral: :\begin \mathrm(r) &= \iint_D 1\ d(x, y)\\ & = \iint_D t\ dt\ d\theta\\ & = \int_0^ \int_0^r t\ dt\ d\theta\\ & = \int_0^ \fracr^2\ d\theta\\ \end Making the substitution u = r\theta,\ du=r\ d\theta converts the integral to :\begin \int_0^ \frac\frac du = \int_0^ \frac r\ du \end which is the same as the above result. The triangle proof can be reformulated as an application of
Green's theorem In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem. Theorem Let be a positively oriented, ...
in flux-divergence form (i.e. a two-dimensional version of the
divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclos ...

divergence theorem
), in a way that avoids all mention of trigonometry and the constant . Consider the
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each attach ...

vector field
\mathbf r=x\mathbf i + y\mathbf j in the plane. So the
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ou ...
of r is equal to two, and hence the area of a disc ''D'' is equal to :A = \frac12\iint_D \operatorname\mathbf r\, dA. By Green's theorem, this is the same as the outward flux of r across the circle bounding ''D'': :A = \frac12\oint_ \mathbf r\cdot\mathbf n\, ds where n is the unit normal and ''ds'' is the arc length measure. For a circle of radius ''R'' centered at the origin, we have , \mathbf r, =R and \mathbf n=\mathbf r/R, so the above equality is :A = \frac12\oint_ \mathbf r\cdot\frac\, ds = \frac\oint_ \,ds. The integral of ''ds'' over the whole circle \partial D is just the arc length, which is its circumference, so this shows that the area ''A'' enclosed by the circle is equal to R/2 times the circumference of the circle. Another proof that uses triangles considers the area enclosed by a circle to be made up of an infinite number of triangles (i.e. the triangles each have an angle of ' at the centre of the circle), each with an area of ' (derived from the expression for the area of a triangle: '). Note that ''≈ '' due to small angle approximation. Through summing the areas of the triangles, the expression for the area of the circle can therefore be found: \begin \mathrm&= \int_0^ \frac r^2 \, d \theta \\ &= \left \frac r^2 \theta \right^\\ &= \pi r^2. \end


Semicircle proof

Note that the area of a semicircle of radius r can be computed by the integral \int_^r \sqrt\,dx. Image:semicircle.svg, frame, A semicircle of radius ''r'' By trigonometric substitution, we substitute x=r \sin\theta , hence dx=r\cos \theta\, d\theta. ::\int_^r \sqrt\,dx :=\int_^\sqrt \cdot r \cos \theta\, d \theta :=2r^2\int_^ \cos ^2 \theta\, d \theta :=\frac. The last step follows since the trigonometric identity \cos(\theta)=\sin(\pi/2-\theta) implies that \cos^2\theta and \sin^2\theta have equal integrals over the interval [0,\pi/2], using integration by substitution. But on the other hand, since \cos^2\theta+\sin^2\theta=1, the sum of the two integrals is the length of that interval, which is \pi/2. Consequently, the integral of \cos^2 \theta is equal to half the length of that interval, which is \pi/4. Therefore, the area of a circle of radius ''r'', which is twice the area of the semi-circle, is equal to 2 \cdot \frac = \pi r^2. This particular proof may appear to beg the question, if the sine and cosine functions involved in the trigonometric substitution are regarded as being defined in relation to circles. However, as noted earlier, it is possible to define sine, cosine, and in a way that is totally independent of trigonometry, in which case the proof is valid by the change of variables formula and Fubini's theorem, assuming the basic properties of sine and cosine (which can also be proved without assuming anything about their relation to circles).


Isoperimetric inequality

The circle is the closed curve of least perimeter that encloses the maximum area. This is known as the isoperimetric inequality, which states that if a rectifiable Jordan curve in the Euclidean plane has perimeter ''C'' and encloses an area ''A'' (by the Jordan curve theorem) then :4\pi A\le C^2. Moreover, equality holds in this inequality if and only if the curve is a circle, in which case A=\pi r^2 and C=2\pi r.


Fast approximation

The calculations Archimedes used to approximate the area numerically were laborious, and he stopped with a polygon of 96 sides. A faster method uses ideas of Willebrord Snell (''Cyclometricus'', 1621), further developed by Christiaan Huygens (''De Circuli Magnitudine Inventa'', 1654), described in .


Archimedes' doubling method

Given a circle, let ''un'' be the
perimeter A perimeter is either a path that encompasses/surrounds/outlines a shape (in two dimensions) or its length (one-dimensional). The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several practical ...
of an inscribed regular ''n-''gon, and let ''Un'' be the perimeter of a circumscribed regular ''n-''gon. Then ''un'' and ''Un'' are lower and upper bounds for the circumference of the circle that become sharper and sharper as ''n'' increases, and their average (''un'' + ''Un'')/2 is an especially good approximation to the circumference. To compute ''un'' and ''Un'' for large ''n'', Archimedes derived the following doubling formulae: :u_ = \sqrt   (geometric mean), and :U_ = \frac    (harmonic mean). Starting from a hexagon, Archimedes doubled ''n'' four times to get a 96-gon, which gave him a good approximation to the circumference of the circle. In modern notation, we can reproduce his computation (and go further) as follows. For a unit circle, an inscribed hexagon has ''u''6 = 6, and a circumscribed hexagon has ''U''6 = 4. Doubling seven times yields : (Here approximates the circumference of the unit circle, which is 2, so approximates .) The last entry of the table has 355113 as one of its Continued fraction#Best rational approximation, best rational approximations; i.e., there is no better approximation among rational numbers with denominator up to 113. The number 355113 is also an excellent approximation to , better than any other rational number with denominator less than 16604.


The Snell–Huygens refinement

Snell proposed (and Huygens proved) a tighter bound than Archimedes': : n \frac < \pi < n \left(2 \sin \frac + \tan \frac\right). This for ''n'' = 48 gives a better approximation (about 3.14159292) than Archimedes' method for ''n'' = 768.


Derivation of Archimedes' doubling formulae

Let one side of an inscribed regular ''n-''gon have length ''sn'' and touch the circle at points A and B. Let A′ be the point opposite A on the circle, so that A′A is a diameter, and A′AB is an inscribed triangle on a diameter. By Thales' theorem, this is a right triangle with right angle at B. Let the length of A′B be ''cn'', which we call the complement of ''sn''; thus ''cn''2+''sn''2 = (2''r'')2. Let C bisect the arc from A to B, and let C′ be the point opposite C on the circle. Thus the length of CA is ''s''2''n'', the length of C′A is ''c''2''n'', and C′CA is itself a right triangle on diameter C′C. Because C bisects the arc from A to B, C′C perpendicularly bisects the chord from A to B, say at P. Triangle C′AP is thus a right triangle, and is similarity (geometry)#Similar triangles, similar to C′CA since they share the angle at C′. Thus all three corresponding sides are in the same proportion; in particular, we have C′A : C′C = C′P : C′A and AP : C′A = CA : C′C. The center of the circle, O, bisects A′A, so we also have triangle OAP similar to A′AB, with OP half the length of A′B. In terms of side lengths, this gives us :\begin c_^2 &= \left( r + \frac c_n \right) 2r \\ c_ &= \frac . \end In the first equation C′P is C′O+OP, length ''r''+12''cn'', and C′C is the diameter, 2''r''. For a unit circle we have the famous doubling equation of Ludolph van Ceulen, : c_ = \sqrt . If we now circumscribe a regular ''n-''gon, with side A″B″ parallel to AB, then OAB and OA″B″ are similar triangles, with A″B″ : AB = OC : OP. Call the circumscribed side ''Sn''; then this is ''Sn'' : ''sn'' = 1 : 12''cn''. (We have again used that OP is half the length of A′B.) Thus we obtain : c_n = 2\frac . Call the inscribed perimeter ''un'' = ''nsn'', and the circumscribed perimeter ''Un'' = ''nSn''. Then combining equations, we have : c_ = \frac = 2 \frac , so that : u_^2 = u_n U_ . This gives a geometric mean equation. We can also deduce : 2 \frac \frac = 2 + 2 \frac , or : \frac = \frac + \frac . This gives a harmonic mean equation.


Dart approximation

When more efficient methods of finding areas are not available, we can resort to "throwing darts". This Monte Carlo method uses the fact that if random samples are taken uniformly scattered across the surface of a square in which a disk resides, the proportion of samples that hit the disk approximates the ratio of the area of the disk to the area of the square. This should be considered a method of last resort for computing the area of a disk (or any shape), as it requires an enormous number of samples to get useful accuracy; an estimate good to 10−''n'' requires about 100''n'' random samples .


Finite rearrangement

We have seen that by partitioning the disk into an infinite number of pieces we can reassemble the pieces into a rectangle. A remarkable fact discovered relatively recently is that we can dissect the disk into a large but ''finite'' number of pieces and then reassemble the pieces into a square of equal area. This is called Tarski's circle-squaring problem. The nature of Laczkovich's proof is such that it proves the existence of such a partition (in fact, of many such partitions) but does not exhibit any particular partition.


Non-Euclidean circles

Circles can be defined in non-Euclidean geometry, and in particular in the hyperbolic plane, hyperbolic and elliptic plane, elliptic planes. For example, the unit sphere S^2(1) is a model for the two-dimensional elliptic plane. It carries an intrinsic metric that arises by measuring geodesic length. The geodesic circles are the parallels in a latitude, geodesic coordinate system. More precisely, fix a point \mathbf z\in S^2(1) that we place at the zenith. Associated to that zenith is a geodesic polar coordinate system (\phi,\theta), 0\le\phi\le\pi, 0\le\theta< 2\pi, where z is the point \phi=0. In these coordinates, the geodesic distance from z to any other point \mathbf x\in S^2(1) having coordinates (\phi,\theta) is the value of \phi at x. A spherical circle is the set of points a geodesic distance ''R'' from the zenith point z. Equivalently, with a fixed embedding into \mathbb R^3, the spherical circle of radius R\le\pi centered at z is the set of x in S^2(1) such that \mathbf x\cdot\mathbf z = \cos R. We can also measure the area of the spherical disk enclosed within a spherical circle, using the intrinsic surface area measure on the sphere. The area of the disk of radius ''R'' is then given by :A = \int_0^\int_0^R \sin(\phi)d\phi\,d\theta = 2\pi(1-\cos R). More generally, if a sphere S^2(\rho) has radius of curvature \rho, then the area of the disk of radius ''R'' is given by :A = 2\pi\rho^2(1-\cos(R/\rho)). Observe that, as an application of L'Hôpital's rule, this tends to the Euclidean area \pi R^2 in the flat limit \rho\to\infty. The hyperbolic case is similar, with the area of a disk of intrinsic radius ''R'' in the (constant curvature -1) hyperbolic plane given by :A = 2\pi(1-\cosh R) where cosh is the hyperbolic cosine. More generally, for the constant curvature -k hyperbolic plane, the answer is :A = 2\pi k^(1-\cosh(kR)). These identities are important for comparison inequalities in geometry. For example, the area enclosed by a circle of radius ''R'' in a flat space is always greater than the area of a spherical circle and smaller than a hyperbolic circle, provided all three circles have the same (intrinsic) radius. That is, :2\pi(1-\cos R) < \pi R^2 < 2\pi(1-\cosh R) for all R>0. Intuitively, this is because the sphere tends to curve back on itself, yielding circles of smaller area than those in the plane, whilst the hyperbolic plane, when immersed into space, develops fringes that produce additional area. It is more generally true that the area of the circle of a fixed radius ''R'' is a strictly decreasing function of the curvature. In all cases, if k is the curvature (constant, positive or negative), then the isoperimetric inequality for a domain with area ''A'' and perimeter ''L'' is :L^2\ge 4\pi A - kA^2 where equality is achieved precisely for the circle.


Generalizations

We can stretch a disk to form an ellipse. Because this stretch is a linear transformation of the plane, it has a distortion factor which will change the area but preserve ''ratios'' of areas. This observation can be used to compute the area of an arbitrary ellipse from the area of a unit circle. Consider the unit circle circumscribed by a square of side length 2. The transformation sends the circle to an ellipse by stretching or shrinking the horizontal and vertical diameters to the major and minor axes of the ellipse. The square gets sent to a rectangle circumscribing the ellipse. The ratio of the area of the circle to the square is /4, which means the ratio of the ellipse to the rectangle is also /4. Suppose ''a'' and ''b'' are the lengths of the major and minor axes of the ellipse. Since the area of the rectangle is ''ab'', the area of the ellipse is ''ab''/4. We can also consider analogous measurements in higher dimensions. For example, we may wish to find the volume inside a sphere. When we have a formula for the surface area, we can use the same kind of "onion" approach we used for the disk.


Bibliography

*
(Originally published by Cambridge University Press, 1897, based on J. L. Heiberg's Greek version.) * *
(Originally ''Grundzüge der Mathematik'', Vandenhoeck & Ruprecht, Göttingen, 1971.) * * * * {{citation , title=Computational Physics , last1=Thijssen , first1=J. M. , pages=273 , publisher=Cambridge University Press , year=2006 , isbn=978-0-521-57588-1


References


External links


Science News on Tarski problem
Area Circles Articles containing proofs de:Kreis#Kreisfläche