Circumference
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Circumference is the distance, or perimeter, around a closed curve.
The circumference of a circle has been found to be directly proportional to its diameter, and is represented by the formula:
where the pi symbol π is a dimensionless constant approximately equal to 3.14159.
A common alternate formula is found by substituting the radius, giving:
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. The circumference of a circle can be calculated from its diameter using the formula:
Or, substituting the diameter for the radius:
where r is the radius and d is the diameter of the circle, and (the Greek letter pi) is defined as the ratio of the circumference of the circle to its diameter (the numerical value of pi is 3.141 592 653 589 793...).
If desired, the above circumference formula can be derived without reference to the definition of π by using some integral calculus, as follows:
The upper half of a circle centered at the origin is the graph of the function , where x runs from to . The circumference (c) of the entire circle can be represented as twice the sum of the lengths of the infinitesimal arcs that make up this half circle. The length of a single infinitesimal part of the arc can be calculated using the Pythagorean formula for the length of the hypotenuse of a right triangle with side lengths and , which gives us .
Thus the circle circumference can be calculated as follows:
The antiderivative needed to solve this definite integral is the arcsine function:
Pi (π) is the ratio of the circumference of a circle to its diameter.
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 are the ellipse's semi-major and semi-minor axes, respectively, and is the ellipse's angular eccentricity,
There are many different approximations for the divided difference, with varying degrees of sophistication and corresponding accuracy.
In comparing the different approximations, the based series expansion is used to find the actual value:
Muir-1883
- Probably the most accurate to its given simplicity is Thomas Muir's:
Ramanujan-1914 (#1,#2)
- Srinivasa Ramanujan introduced two different approximations, both from 1914
- The second equation is demonstratively 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
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