Area of a circle
Encyclopedia
The area of a circle is πr2 when the circle has radius
r. Here the symbol π (Greek
letter pi
) denotes, as usual, the constant ratio of the circumference
of a circle to its diameter
. It is easy to deduce the area
of a disk
from basic principles: the area of a regular polygon
is half its apothem
times its perimeter, and a regular polygon becomes a circle as the number of sides increases, so the area of a disk is half its radius times its circumference (i.e. r × 2πr).
Modern mathematics can obtain the area using the methods of integral calculus or its more sophisticated offspring, real analysis
. However, in Ancient Greece
the great mathematician Archimedes
used the tools of Euclidean geometry
to show that the area inside a circle is equal to that of a right triangle
whose base has the length of the circle's circumference and whose height equals the circle's radius in his book Measurement of a Circle
. The circumference is 2πr, and the area of a triangle is half the base times the height, yielding the area πr2 for the disk.
is half its perimeter times the apothem
. As the number of sides of the regular polygon increases, it becomes identical to a circle, and the apothem becomes identical to the radius. Therefore, the area of a circle is half its circumference times the radius.
s in an essential way.
Suppose the circle area, C, may be greater than the triangle area, T = 1⁄2cr. 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, G4, 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, G8. 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.
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 1⁄2nhs. But since h < r and ns < c, the polygon area must be less than the triangle area, 1⁄2cr, a contradiction. Therefore our supposition that C might be greater than T must be wrong.
Suppose the circle area may be less than the triangle area. 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, G4, 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.
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 of the circle is precisely the same as the area of the triangle. This concludes the proof.
Following Satō Moshun and Leonardo da Vinci
, 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
, with the hexagon sides making two opposite edges, one of which is the base, s. Two radial edges form slanted sides, and the height is h (as in the Archimedes proof). In fact, we can assemble all the triangles into one big parallelogram by putting successive pairs next to each other. The same is true if we increase to eight sides and so on. For a polygon with 2n 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 πr and height r.
Using calculus, we can sum the area incrementally, partitioning the disk into thin concentric rings like the layers of an onion
. 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 2πt dt, the circumferential length of the ring times its infinitesimal width (you can approach this ring by a rectangle with width=2πt and height=dt). This gives an elementary integral for a disk of radius r.
Given a circle, let un be the perimeter length of an inscribed regular n-gon, and let Un be the perimeter length of a circumscribed regular n-gon. Then we have the following doubling formulae.
(geometric mean
)
(harmonic mean
)
Archimedes doubled a hexagon four times to get a 96-gon. For a unit circle, an inscribed hexagon has u6 = 6, and a circumscribed hexagon has U6 = 4√3. We have the luxury of decimal notation and our two equations, so we can quickly double seven times:
A best rational approximation to the last average is 355⁄113, which is an excellent value for π. But Snell proposes (and Huygens proves) a tighter bound than Archimedes.
Thus we could get the same approximation, with decimal value about 3.14159292, from a 48-gon.
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 cn2+sn2 = (2r)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 s2n, the length of C′A is c2n, 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
In the first equation C′P is C′O+OP, length r+1⁄2cn, and C′C is the diameter, 2r. For a unit circle we have the famous doubling equation of Ludolph van Ceulen
,
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 : 1⁄2cn. (We have again used that OP is half the length of A′B.) Thus we obtain
Call the inscribed perimeter un = nsn, and the circumscribed perimenter Un = nSn. Then combining equations, we have
so that
This gives a geometric mean
equation.
We can also deduce
or
This gives a harmonic mean
equation.
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 100n random samples .
. 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.
. 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 π/4, which means the ratio of the ellipse to the rectangle is also π/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 πab/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.
Finding the area of this triangle will give the area of circle
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
r. Here the symbol π (Greek
Greek alphabet
The Greek alphabet is the script that has been used to write the Greek language since at least 730 BC . The alphabet in its classical and modern form consists of 24 letters ordered in sequence from alpha to omega...
letter pi
Pi (letter)
Pi is the sixteenth letter of the Greek alphabet, representing . In the system of Greek numerals it has a value of 80. Letters that arose from pi include Cyrillic Pe , Coptic pi , and Gothic pairthra .The upper-case letter Π is used as a symbol for:...
) denotes, as usual, the constant ratio of the circumference
Circumference
The circumference is the distance around a closed curve. Circumference is a special perimeter.-Circumference of a circle:The circumference of a circle is the length around it....
of a circle to its diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...
. It is easy to deduce the area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...
of a disk
Disk (mathematics)
In geometry, a disk is the region in a plane bounded by a circle.A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary...
from basic principles: the area of a regular polygon
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...
is half its apothem
Apothem
The apothem 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 one of its sides. The word "apothem" can also refer to the length of that line segment. Regular polygons...
times its perimeter, and a regular polygon becomes a circle as the number of sides increases, so the area of a disk is half its radius times its circumference (i.e. r × 2πr).
Modern mathematics can obtain the area using the methods of integral calculus or its more sophisticated offspring, real analysis
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
. However, in Ancient Greece
Ancient Greece
Ancient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...
the great mathematician Archimedes
Archimedes
Archimedes of Syracuse 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. Among his advances in physics are the foundations of hydrostatics, statics and an...
used the tools of Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the 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, and deducing many other propositions from these...
to show that the area inside a circle is equal to that of a right triangle
Right triangle
A right triangle or right-angled triangle is a triangle in which one angle is a right angle . The relation between the sides and angles of a right triangle is the basis for trigonometry.-Terminology:The side opposite the right angle is called the hypotenuse...
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
Measurement of a Circle is a treatise that consists of three propositions by Archimedes. The treatise is only a fraction of what was a longer work.-Proposition one:Proposition one states:...
. The circumference is 2πr, and the area of a triangle is half the base times the height, yielding the area πr2 for the disk.
Using polygons
The area of a regular polygonRegular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...
is half its perimeter times the apothem
Apothem
The apothem 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 one of its sides. The word "apothem" can also refer to the length of that line segment. Regular polygons...
. As the number of sides of the regular polygon increases, it becomes identical to a circle, and the apothem becomes identical to the radius. Therefore, the area of a circle is half its circumference times the radius.
Archimedes's proof
Following , compare 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 polygonRegular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...
s in an essential way.
Not greater
Inscribe
right|thumb|An inscribed triangle of a circleIn 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 figure G" means precisely the same thing as "figure G is circumscribed about...
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, G4, 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, G8. 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.
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 1⁄2nhs. But since h < r and ns < c, the polygon area must be less than the triangle area, 1⁄2cr, a contradiction. Therefore our supposition that C might be greater than T must be wrong.
Not less
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 of the circle is precisely the same as the area of the triangle. This concludes the proof.
Rearrangement proof
Leonardo da Vinci
Leonardo di ser Piero da Vinci was an Italian Renaissance polymath: painter, sculptor, architect, musician, scientist, mathematician, engineer, inventor, anatomist, geologist, cartographer, botanist and writer whose genius, perhaps more than that of any other figure, epitomized the Renaissance...
, 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
Parallelogram
In Euclidean geometry, a parallelogram is a convex 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 measure...
, with the hexagon sides making two opposite edges, one of which is the base, s. Two radial edges form slanted sides, and the height is h (as in the Archimedes proof). In fact, we can assemble all the triangles into one big parallelogram by putting successive pairs next to each other. The same is true if we increase to eight sides and so on. For a polygon with 2n 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 πr and height r.
| polygon | | parallelogram | |||||||
|---|---|---|---|---|---|---|---|---|
| n | side | base | height | area | ||||
| 4 | 1.4142136 | 2.8284271 | 0.7071068 | 2.0000000 | ||||
| 6 | 1.0000000 | 3.0000000 | 0.8660254 | 2.5980762 | ||||
| 8 | 0.7653669 | 3.0614675 | 0.9238795 | 2.8284271 | ||||
| 10 | 0.6180340 | 3.0901699 | 0.9510565 | 2.9389263 | ||||
| 12 | 0.5176381 | 3.1058285 | 0.9659258 | 3.0000000 | ||||
| 14 | 0.4450419 | 3.1152931 | 0.9749279 | 3.0371862 | ||||
| 16 | 0.3901806 | 3.1214452 | 0.9807853 | 3.0614675 | ||||
| 96 | 0.0654382 | 3.1410320 | 0.9994646 | 3.1393502 | ||||
| ∞ | 1/∞ | π | 1 | π | ||||
Onion proof
Onion
The onion , also known as the bulb onion, common onion and garden onion, is the most widely cultivated species of the genus Allium. The genus Allium also contains a number of other species variously referred to as onions and cultivated for food, such as the Japanese bunching onion The onion...
. This is the method of shell integration
Shell integration
Shell integration is a means of calculating the volume of a solid of revolution, when integrating along an axis perpendicular to the axis of revolution.It makes use of the so-called "representative cylinder"...
in two dimensions. For an infinitesimally thin ring of the "onion" of radius t, the accumulated area is 2πt dt, the circumferential length of the ring times its infinitesimal width (you can approach this ring by a rectangle with width=2πt and height=dt). This gives an elementary integral for a disk of radius 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) followed up by Christiaan Huygens (De Circuli Magnitudine Inventa, 1654), described in .Given a circle, let un be the perimeter length of an inscribed regular n-gon, and let Un be the perimeter length of a circumscribed regular n-gon. Then we have the following doubling formulae.
(geometric meanGeometric mean
The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...
)
(harmonic meanHarmonic mean
In mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....
)
Archimedes doubled a hexagon four times to get a 96-gon. For a unit circle, an inscribed hexagon has u6 = 6, and a circumscribed hexagon has U6 = 4√3. We have the luxury of decimal notation and our two equations, so we can quickly double seven times:
| k | n | un | Un | (un + Un)/4 | ||||
|---|---|---|---|---|---|---|---|---|
| 0 | 6 | 6.0000000 | 6.9282032 | 3.2320508 | ||||
| 1 | 12 | 6.2116571 | 6.4307806 | 3.1606094 | ||||
| 2 | 24 | 6.2652572 | 6.3193199 | 3.1461443 | ||||
| 3 | 48 | 6.2787004 | 6.2921724 | 3.1427182 | ||||
| 4 | 96 | 6.2820639 | 6.2854292 | 3.1418733 | ||||
| 5 | 192 | 6.2829049 | 6.2837461 | 3.1416628 | ||||
| 6 | 384 | 6.2831152 | 6.2833255 | 3.1416102 | ||||
| 7 | 768 | 6.2831678 | 6.2832204 | 3.1415970 |
A best rational approximation to the last average is 355⁄113, which is an excellent value for π. But Snell proposes (and Huygens proves) a tighter bound than Archimedes.
Thus we could get the same approximation, with decimal value about 3.14159292, from a 48-gon.
Derivation
Thales' theorem
In geometry, Thales' theorem states that if A, B and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. Thales' theorem is a special case of the inscribed angle 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 cn2+sn2 = (2r)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 s2n, the length of C′A is c2n, 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
In the first equation C′P is C′O+OP, length r+1⁄2cn, and C′C is the diameter, 2r. For a unit circle we have the famous doubling equation of Ludolph van Ceulen
Ludolph van Ceulen
Ludolph van Ceulen was a German / Dutch mathematician from Hildesheim. He emigrated to the Netherlands....
,
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 : 1⁄2cn. (We have again used that OP is half the length of A′B.) Thus we obtain
Call the inscribed perimeter un = nsn, and the circumscribed perimenter Un = nSn. Then combining equations, we have
so that
This gives a geometric mean
Geometric mean
The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...
equation.
We can also deduce
or
This gives a harmonic mean
Harmonic mean
In mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....
equation.
Dart approximation
Monte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
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 100n 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 problemTarski's circle-squaring problem
Tarski's circle-squaring problem is the challenge, posed by Alfred Tarski in 1925, to take a disc in the plane, cut it into finitely many pieces, and reassemble the pieces so as to get a square of equal area. This was proven to be possible by Miklós Laczkovich in 1990; the decomposition makes heavy...
. 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.
Generalizations
We can stretch a disk to form an ellipseEllipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
. Because this stretch is a linear transformation
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
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 π/4, which means the ratio of the ellipse to the rectangle is also π/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 πab/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.
Triangle method
This approach is a slight modification of onion proof. Consider unwrapping the concentric circles to straight strips. This will form a right angled triangle with r as its height and 2πr (being the outer slice of onion) as its base.Finding the area of this triangle will give the area of circle
External links
- Area enclosed by a circle (with interactive animation)
- Science News on Tarski problem