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Circumference of an Ellipse

  1. Dec 18, 2003 #1
    is there a formula to find the circumference of an ellipse?
  2. jcsd
  3. Dec 18, 2003 #2


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    You can approximate it. An exact expression for the perimeter of an ellipse is
    [itex] P = 4a\int_0^{\frac{\pi}{2}}\sqrt{1-e^2\sin^2{t}}dt [/itex]
    where a is the semi-major axis, the eccentricity [itex] e = \frac{\sqrt{a^2-b^2}}{a} [/itex], and b is the semi-minor axis. This is found writing the equation of the ellipse in parametric form and using the arc-length formula. You can compute the integral numerically or write an approximation using a series expansion.

    Here is a website with a number of approximations you can try out:

    http://astronomy.swin.edu.au/~pbourke/geometry/ellipsecirc/ [Broken]
    Last edited by a moderator: May 1, 2017
  4. Dec 18, 2003 #3
    Length of the curve is given by

    [tex] s=\int_a^b \sqrt{1+(\frac{dy}{dx})^2}dx[/tex]
    where y=f(x) and x=a,x=b
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