In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, the
area
Area is the quantity that expresses the extent of a region on the plane or on a curved 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 ope ...
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 con ...
of
radius
In classical geometry, a radius ( : radii) 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 ...
is . Here the Greek letter
represents the
constant 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 t ...
of any circle to its
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center 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 fo ...
, approximately equal to 3.14159.
One method of deriving this formula, which originated with
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 direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
s. The area of a regular polygon is half its
perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pr ...
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'' 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
In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
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 conv ...
. However, the area of a disk was studied by the
Ancient Greeks
Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cult ...
.
Eudoxus of Cnidus in the fifth century B.C. had found that the area of a disk is proportional to its radius squared.
Archimedes used the tools of
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach 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), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right a ...
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
Prop ...
''. 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 misadve ...
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 p ...
,
[.] but did not identify the
constant of proportionality
In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constan ...
.
Historical arguments
A variety of arguments have been advanced historically to establish the equation
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 are ...
, 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 direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
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 direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
s in the same way.
Not greater
Suppose that the area ''C'' enclosed by the circle is greater than the area ''T'' =
1⁄
2''cr'' of the triangle. Let ''E'' denote the excess amount.
Inscribe
{{unreferenced, date=August 2012
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 say that "figure F is inscribed in figur ...
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, ''G
n'', is less than ''E''. Now the area of the inscribed polygon, ''P
n'' = ''C'' − ''G
n'', must be greater than that of the triangle.
:
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
1⁄
2''nhs''. But since ''h'' < ''r'' and ''ns'' < ''c'', the polygon area must be less than the triangle area,
1⁄
2''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, ''P
n'', must be less than ''T''.
:
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 di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially res ...
, 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. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A '' regular hexagon'' has ...
. 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, 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:
:
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 function or, equivalently, the half-period of the
sine (or cosine) function. The cosine function can be defined either as a
power series, or as the solution of a certain
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
. 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 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 wit ...
:
The integral appearing on the right is an
abelian integral In 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, wh ...
whose value is a half-period of the
sine function, equal to . Thus
is seen to be true as a theorem.
Several of the arguments that follow use only concepts from elementary calculus to reproduce the formula
, 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'' L., from Latin ''cepa'' meaning "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 onio ...
. 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 whi ...
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''.
:
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 ...
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 th ...
as follows:
:
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
by
. Note ρ is a
Lipschitz function 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 "direction and rate of fastest increase". If the gr ...
is a unit vector
(
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
in
. We will show that
, where
is the two-dimensional Lebesgue measure in
. We shall assume that the one-dimensional
Hausdorff measure of the circle
is
, the circumference of the circle of radius ''r''. (This can be taken as the definition of circumference.) Then, by the coarea formula,
:
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
:
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
. By summing up (integrating) all of the areas of these triangles, we arrive at the formula for the circle's area:
:
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 ''d''
A time s ...
and using a change of variables in the above iterated integral:
:
Making the substitution
converts the integral to
:
which is the same as the above result.
The triangle proof can be reformulated as an application of
Green's theorem 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 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 t ...
of r is equal to two, and hence the area of a disc ''D'' is equal to
:
By Green's theorem, this is the same as the outward flux of r across the circle bounding ''D'':
:
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
and
, so the above equality is
:
The integral of ''ds'' over the whole circle
is just the arc length, which is its circumference, so this shows that the area ''A'' enclosed by the circle is equal to
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:
Semicircle proof
Note that the area of a semicircle of radius ''r'' can be computed by the integral
.
By
trigonometric substitution
In mathematics, trigonometric substitution is the replacement of trigonometric functions for other expressions. In calculus, trigonometric substitution is a technique for evaluating integrals. Moreover, one may use the trigonometric identities ...
, we substitute
, hence
The last step follows since the trigonometric identity
implies that
and
have equal integrals over the interval