Circumference
Encyclopedia
The circumference is the distance around a closed curve. Circumference is a special perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length  it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference. Practical uses :Calculating...
.
Circumference of a circle
The circumference of a circle is the length around it.The circumference of a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
can be calculated from 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...
using the formula:
Or, substituting the radius
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...
for the diameter:
where is the radius
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...
and is the diameter of the circle, and the Greek letter is defined as the ratio of the circumference of the circle to its diameter. The numerical value of is 3.141 592 653 589 793....
Circumference of an ellipse
The circumference of 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...
is more problematic, as the exact solution requires finding the complete elliptic integral of the second kind. This can be achieved either via numerical integration
Numerical integration
In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of...
(the best type being Gaussian quadrature
Gaussian quadrature
In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration....
) or by one of many binomial series
Binomial series
In mathematics, the binomial series is the Taylor series at x = 0 of the function f given by f = α, where is an arbitrary complex number...
expansions.
Where , are the ellipse's semimajor
Semimajor axis
The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...
and semiminor
Semiminor axis
In geometry, the semiminor axis is a line segment associated with most conic sections . One end of the segment is the center of the conic section, and it is at right angles with the semimajor axis...
axes, respectively, and is the ellipse's angular eccentricity
Angular eccentricity
In the study of ellipses and related geometry, various parameters in the distortion of a circle into an ellipse are identified and employed: Aspect ratio, flattening and eccentricity....
,
There are many different approximation
Approximation
An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.Approximations may be used because...
s for the divided difference
Difference quotient
The primary vehicle of calculus and other higher mathematics is the function. Its "input value" is its argument, usually a point expressible on a graph...
, with varying degrees of sophistication and corresponding accuracy.
In comparing the different approximations, the (also known as "", the third flattening
Flattening
The flattening, ellipticity, or oblateness of an oblate spheroid is a measure of the "squashing" of the spheroid's pole, towards its equator...
of the ellipse) based series expansion is used to find the actual value:
More generally, the arc length
Arc length
Determining the length of an irregular arc segment is also called rectification of a curve. Historically, many methods were used for specific curves...
of a portion of the circumference, as a function of the angle subtended, is given by an incomplete elliptic integral
Elliptic integral
In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler...
.
The inverse function
Inverse function
In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...
, the angle subtended as a function of the arc length, is given by the elliptic functions.
Muir1883
 Probably the most accurate to its given simplicity is Thomas Muir'sThomas Muir (mathematician)Sir Thomas Muir FRS was a Scottish mathematician, remembered as an authority on determinants. He was born in Stonebyres in South Lanarkshire, and brought up in the small town of Biggar. At the University of Glasgow he changed his studies from classics to mathematics after advice from the future...
:
Ramanujan1914 (#1,#2)
 Srinivasa RamanujanSrinivasa RamanujanSrīnivāsa Aiyangār Rāmānujan FRS, better known as Srinivasa Iyengar Ramanujan was a Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series and continued fractions...
introduced two different approximations, both from 1914

 The second equation is demonstrably by far the better of the two, and may be the most accurate approximation known.
Letting = 10000 and = ×cos{ } , results with different ellipticities can be found and compared:
b Pr Ramanujan#2 Ramanujan#1 Muir 9975 9987.50391 11393 9987.50391 11393 9987.50391 11393 9987.50391 11389 9966 9983.00723 73047 9983.00723 73047 9983.00723 73047 9983.00723 73034 9950 9975.01566 41666 9975.01566 41666 9975.01566 41666 9975.01566 41604 9900 9950.06281 41695 9950.06281 41695 9950.06281 41695 9950.06281 40704 9000 9506.58008 71725 9506.58008 71725 9506.58008 67774 9506.57894 84209 8000 9027.79927 77219 9027.79927 77219 9027.79924 43886 9027.77786 62561 7500 8794.70009 24247 8794.70009 24240 8794.69994 52888 8794.64324 65132 6667 8417.02535 37669 8417.02535 37460 8417.02428 62059 8416.81780 56370 5000 7709.82212 59502 7709.82212 24348 7709.80054 22510 7708.38853 77837 3333 7090.18347 61693 7090.18324 21686 7089.94281 35586 7083.80287 96714 2500 6826.49114 72168 6826.48944 11189 6825.75998 22882 6814.20222 31205 1000 6468.01579 36089 6467.94103 84016 6462.57005 00576 6431.72229 28418 100 6367.94576 97209 6366.42397 74408 6346.16560 81001 6303.80428 66621 10 6366.22253 29150 6363.81341 42880 6340.31989 06242 6299.73805 61141 1 6366.19804 50617 6363.65301 06191 6339.80266 34498 6299.60944 92105 iota IotaIota is the ninth letter of the Greek alphabet. In the system of Greek numerals it has a value of 10. It was derived from the Phoenician letter Yodh . Letters that arose from this letter include the Roman I and J and the Cyrillic І , Yi , Je , and iotified letters .Iota represents...6366.19772 36758 6363.63636 36364 6339.74596 21556 6299.60524 94744
Circumference of a graph
In graph theoryGraph theoryIn mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
the circumference of a graphGraph (mathematics)In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
refers to the longest cycleCycle (graph theory)In graph theory, the term cycle may refer to a closed path. If repeated vertices are allowed, it is more often called a closed walk. If the path is a simple path, with no repeated vertices or edges other than the starting and ending vertices, it may also be called a simple cycle, circuit, circle,...
contained in that graph.
External links
 Numericana  Circumference of an ellipse
 Circumference of a circle With interactive applet and animation

 Srinivasa Ramanujan