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Encyclopedia > Circumference

The circumference is the distance around a closed curve. Circumference is a kind of perimeter. The perimeter is the distance around a given two-dimensional object. ...

Circumference = π × diameter
Circumference = π × diameter

Contents

Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... When a circles diameter is 1, its circumference is π. Pi or π is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ...

Circle

The circumference of a circle can be calculated from its diameter using the formula: Circle illustration This article is about the shape and mathematical concept of circle. ... DIAMETER is a computer networking protocol for AAA (Authentication, Authorization and Accounting). ...

c=picdot{d}.,!

Or, substituting the radius for the diameter:

c=2picdot{r}=picdot{2r},,!

where r is the radius and d is the diameter of the circle, and π (the Greek letter pi) is the constant 3.141 592 653 589 793... Remote Authentication Dial In User Service (RADIUS) is an AAA (authentication, authorization and accounting) protocol for applications such as network access or IP mobility. ... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ... A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ...


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. For other uses, see Ellipse (disambiguation). ... The complete elliptic integral of the second kind E may be defined as or It is a special case of the incomplete elliptic integral of the second kind: Category: ... Numerical Integration with the Monte Carlo method: Nodes are random equally distributed. ... 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. ... In mathematics, the binomial series generalizes the purely algebraic binomial theorem. ...


Where a,b are the ellipse's semi-major and semi-minor axes, respectively, and o!varepsilon,! is the ellipse's angular eccentricity, The semi-major axis of an ellipse In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae. ... In geometry, the semi-minor axis (also semiminor axis) applies to ellipses and hyperbolas. ... 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. ...


o!varepsilon=arccos!left(frac{b}{a}right)=2arctan!left(!sqrt{frac{a-b}{a+b}},right);,!


begin{align}mbox{E2}left[0,90^circright]&= mbox{Integral}'smbox{ divided difference}; Pr&=atimesmbox{E2}left[0,90^circright] quad(mbox{perimetric radius}); c&=2pitimes Pr.end{align},!


There are many different approximations for the mbox{E2}left[0,90^circright] divided difference, with varying degrees of sophistication and corresponding accuracy. It has been suggested that this article or section be merged with estimation. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ...


In comparing the different approximations, the tan!left(frac{o!varepsilon}{2}right)^2,! based series expansion is used to find the actual value:


begin{align}mbox{E2}left[0,90^circright] &=cos!left(frac{o!varepsilon}{2}right)^2 frac{1}{UT}sum_{TN=1}^{UT=infty}{.5choose{}TN}^2tan!left(frac{o!varepsilon}{2}right)^{4TN}, &=cos!left(frac{o!varepsilon}{2}right)^2Bigg(1+frac{1}{4}tan!left(frac{o!varepsilon}{2}right)^4 +frac{1}{64}tan!left(frac{o!varepsilon}{2}right)^8 &qquadqquadqquad;,+frac{1}{256}tan!left(frac{o!varepsilon}{2}right)^{12} +frac{25}{16384}tan!left(frac{o!varepsilon}{2}right)^{16} +...Bigg);end{align},!


Muir-1883

Probably the most accurate to its given simplicity is Thomas Muir's:
begin{align}Pr &approxleft(frac{a^{1.5}+b^{1.5}}{2}right)^frac{1}{1.5}=aleft(frac{1+cos!left(o!varepsilonright)^{1.5}}{2}right)^frac{1}{1.5}, &quadapprox{a}timescos!left(frac{o!varepsilon}{2}right)^2left(1+frac{1}{4}tan!left(frac{o!varepsilon}{2}right)^4right);end{align},!

Sir Thomas Muir (25 August 1844-21 March 1934) was a Scottish mathematician, remembered as an authority on determinants. ...

Ramanujan-1914 (#1,#2)

Srinivasa Ramanujan introduced two different approximations, both from 1914
begin{align}1.;Pr&approxpiBig(3(a+b)-sqrt{big(3a+bbig)big(a+3bbig)}Big), &quad=pi{a}bigg(6cos!left(frac{o!varepsilon}{2}right)^2sqrt{big(3+cos!left(o!varepsilonright)big)big(1+3cos!left(o!varepsilonright)big)}bigg);end{align},!
begin{align}2.;Pr&approxfrac{1}{2}Big(a+bBig)Bigg(1+frac{3big(frac{a-b}{a+b}big)^2}{10+sqrt{4-3big(frac{a-b}{a+b}big)^2}}Bigg); &quad=atimescos!left(frac{o!varepsilon}{2}right)^2Bigg(1+frac{3tan!big(frac{o!varepsilon}{2}big)^4}{10+sqrt{4-3tan!big(frac{o!varepsilon}{2}big)^4}}Bigg);end{align},!
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: Srinivasa Ramanujan Iyengar (Tamil: ) (22 December 1887 – 26 April 1920) was an Indian mathematician who is widely regarded as one of the greatest mathematical minds in recent history. ...

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

External links

Look up circumference in Wiktionary, the free dictionary.

  Results from FactBites:
 
Circumference (530 words)
is the ratio of the circumference of a circle to the diameter.
The circumference of a circle is 15.7 centimeters.
The diameter of a nickel is 2 centimeters.
SparkNotes: Geometric Measurements: Circumference (372 words)
The circumference of a circle is the length of the curve that encloses that circle.
In fact, the circumference is dependent solely on the radius of a circle: circumference equals 2πr, where r denotes the length of the radius.
Thus, the circumference of a circle is the length of the 360 degree arc of that circle.
  More results at FactBites »


 

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