area of a disk
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geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, the
area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
enclosed by a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
of
radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...
is . Here, the Greek letter represents the constant ratio of the
circumference In geometry, the circumference () is the perimeter of a circle or ellipse. The circumference is the arc length of the circle, as if it were opened up and straightened out to a line segment. More generally, the perimeter is the curve length arou ...
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 (geometry), chord of the circle. Both definitions a ...
, approximately equal to 3.14159. One method of deriving this formula, which originated with
Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
, 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 polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...
s with an increasing number of sides. The area of a regular polygon is half its
perimeter A perimeter is the length of a closed boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimet ...
multiplied by the distance from its center to its sides, and because the sequence tends to a circle, the corresponding formula–that the area is half the
circumference In geometry, the circumference () is the perimeter of a circle or ellipse. The circumference is the arc length of the circle, as if it were opened up and straightened out to a line segment. More generally, the perimeter is the curve length arou ...
times the radius–namely, , holds for a circle.


Terminology

Although often referred to as the area of a circle in informal contexts, strictly speaking, the term disk refers to the interior region 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 ...
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 or its more sophisticated offspring,
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include co ...
. However, the area of a disk was studied by the
Ancient Greeks Ancient Greece () was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity (), that comprised a loose collection of culturally and linguistically re ...
.
Eudoxus of Cnidus Eudoxus of Cnidus (; , ''Eúdoxos ho Knídios''; ) was an Ancient Greece, ancient Greek Ancient Greek astronomy, astronomer, Greek mathematics, mathematician, doctor, and lawmaker. He was a student of Archytas and Plato. All of his original work ...
in the fifth century B.C. had found that the area of a disk is proportional to its radius squared.
Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
used the tools of
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
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 in his book '' Measurement of a Circle''. 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 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,. but did not identify the constant of proportionality.


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 (one at a time) whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the differ ...
, one of the earliest uses of the mathematical concept of a limit, as well as the origin of Archimedes' axiom which remains part of the standard analytical treatment of the real number system. 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 polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...
is half its perimeter times the apothem. 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 polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...
s in the same way.


Not greater

Suppose that the area ''C'' enclosed by the circle is greater than the area ''T'' = ''cr''/2 of the triangle. Let ''E'' denote the excess amount. Inscribe 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 is ''ns'', which is less than the circle circumference. The polygon area consists of ''n'' equal triangles with height ''h'' and base ''s'', thus equals ''nhs''/2. But since ''h'' < ''r'' and ''ns'' < ''c'', the polygon area must be less than the triangle area, ''cr''/2, 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 ,
Nicholas of Cusa Nicholas of Cusa (1401 – 11 August 1464), also referred to as Nicholas of Kues and Nicolaus Cusanus (), was a German Catholic bishop and polymath active as a philosopher, theologian, jurist, mathematician, and astronomer. One of the first Ger ...
and
Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 1452 - 2 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially rested o ...
, we can use inscribed regular polygons in a different way. Suppose we inscribe a 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 polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram a ...
, 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 (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, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that ...
function or, equivalently, the half-period of the
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite th ...
(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) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
, or as the solution of a certain differential equation. 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 by means of the
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
:C = 2\int_^R \frac = 2R\int_^1\frac. The integral appearing on the right is an abelian integral whose value is a half-period of the
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite th ...
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 An onion (''Allium cepa'' , from Latin ), 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 which was classifie ...
. This is the method of 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 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 specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are *the point's distance from a reference ...
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 formula. Define a function \rho:\mathbb R^2\to\mathbb R by \rho(x,y)=\sqrt. Note ρ is a Lipschitz function whose
gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
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 ...
). 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 assi ...
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 \\ pt &= \frac \cdot 2 \pi r \cdot r \\ pt &= \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 \\ pt &= \left \frac r u \right^\\ pt &= \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 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 :\int_0^ \frac \frac du = \int_0^ \frac r\ du 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 (surface in \R^2) bounded by . It is the two-dimensional special case of Stokes' theorem (surface in \R^3) ...
in flux-divergence form (i.e. a two-dimensional version of the 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 space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
\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 rate that the vector field alters the volume in an infinitesimal neighborhood of each point. (In 2D this "volume" refers to ...
of r is equal to two, and hence the area of a disc ''D'' is equal to :A = \frac 1 2 \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 = \frac 1 2 \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 = \frac 1 2\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 center 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. By trigonometric substitution, we substitute x=r \sin\theta , hence dx=r\cos \theta\, d\theta. \begin\int_^r \sqrt\,dx &=\int_^ \sqrt \cdot r \cos \theta\, d \theta \\ pt&=2r^2\int_^ \cos ^2 \theta\, d \theta \\ pt&=\frac. \end 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 ,\pi/2/math>, using
integration by substitution In calculus, integration by substitution, also known as ''u''-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and c ...
. 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 Christiaan Huygens, Halen, Lord of Zeelhem, ( , ; ; also spelled Huyghens; ; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor who is regarded as a key figure in the Scientific Revolution ...
(''De Circuli Magnitudine Inventa'', 1654), described in .


Archimedes' doubling method

Given a circle, let ''un'' be the
perimeter A perimeter is the length of a closed boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimet ...
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 In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
), and :U_ = \frac    (
harmonic mean In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means. It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments. The 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 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 , attributed to Chinese mathematician Zu Chongzhi, who named it Milü. This approximation is better than any other rational number with denominator less than 16,604.


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 In geometry, Thales's theorem states that if , , and are distinct points on a circle where the line is a diameter, the angle is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as pa ...
, 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 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'' + ''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 In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
equation. We can also deduce : 2 \frac \frac = 2 + 2 \frac , or : \frac = \frac + \frac . This gives a
harmonic mean In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means. It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments. The 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 Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be ...
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 and elliptic planes. For example, the
unit sphere In mathematics, a unit sphere is a sphere of unit radius: the locus (mathematics), set of points at Euclidean distance 1 from some center (geometry), center point in three-dimensional space. More generally, the ''unit -sphere'' is an n-sphere, -s ...
S^2(1) is a model for the two-dimensional elliptic plane. It carries an intrinsic metric that arises by measuring
geodesic In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...
length. The geodesic circles are the parallels in a 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 (\varphi,\theta), 0\le\varphi\le\pi, 0\le\theta< 2\pi, where z is the point \varphi=0. In these coordinates, the geodesic distance from z to any other point \mathbf x\in S^2(1) having coordinates (\varphi,\theta) is the value of \varphi 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(\varphi) \, d\varphi\,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 In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
. Because this stretch is a
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
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.


See also

* Area-equivalent radius * Area of a triangle


References


Bibliography

*
(Originally published by
Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...
, 1897, based on J. L. Heiberg's Greek version.) * *
(Originally ''Grundzüge der Mathematik'', Vandenhoeck & Ruprecht, Göttingen, 1971.) * * * *


External links


Science News on Tarski problem
{{Webarchive, url=https://web.archive.org/web/20080413114409/http://www.sciencenews.org/articles/20041030/mathtrek.asp , date=2008-04-13 Area Circles Articles containing proofs de:Kreis#Kreisfläche