 Circumference

Part of a series of articles on the mathematical constant π Uses Area of disk · Circumference
Use in other formulaeProperties Irrationality · Transcendence
Less than 22/7Value Approximations · Memorization People Archimedes · Liu Hui · Zu Chongzhi
Madhava of Sangamagrama
William Jones · John Machin
John Wrench · Ludolph van CeulenHistory Chronology · Book In culture Legislation · Holiday Related topics Squaring the circle · Basel problem
Tau (τ) · Other topics related to πThe circumference is the distance around a closed curve. Circumference is a special perimeter.
Contents
Circumference of a circle
The circumference of a circle is the length around it. The circumference of a circle can be calculated from its diameter using the formula:
Or, substituting the radius for the diameter:
where r is the radius and d is the diameter of the circle, and the Greek letter π is defined as the ratio of the circumference of the circle to its diameter. The numerical value of π is 3.141 592 653 589 793....
Circumference of an ellipse
The circumference of an ellipse 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 (the best type being Gaussian quadrature) or by one of many binomial series expansions.
Where a,b are the ellipse's semimajor and semiminor axes, respectively, and α is the ellipse's angular eccentricity,
There are many different approximations for the E2 [0,90°] divided difference, with varying degrees of sophistication and corresponding accuracy.
In comparing the different approximations, the (also known as "n", the third flattening of the ellipse) based series expansion is used to find the actual value:
More generally, the arc length of a portion of the circumference, as a function of the angle subtended, is given by an incomplete elliptic integral.
The inverse function, 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's:
Ramanujan1914 (#1,#2)
 Srinivasa Ramanujan 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 a = 10000 and b = a×cos{oε}, 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 6366.19772 36758 6363.63636 36364 6339.74596 21556 6299.60524 94744 Circumference of a graph
In graph theory the circumference of a graph refers to the longest cycle contained in that graph.
External links
 Numericana  Circumference of an ellipse
 Circumference of a circle With interactive applet and animation
Categories: Geometric measurement
 Probably the most accurate to its given simplicity is Thomas Muir's:
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