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is there a formula to find the circumference of an ellipse?

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- Thread starter tandoorichicken
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- #1

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is there a formula to find the circumference of an ellipse?

- #2

jamesrc

Science Advisor

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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]

[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]

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- #3

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[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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