# Circumference of an Ellipse

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

jamesrc
Gold Member
You can approximate it. An exact expression for the perimeter of an ellipse is
$P = 4a\int_0^{\frac{\pi}{2}}\sqrt{1-e^2\sin^2{t}}dt$
where a is the semi-major axis, the eccentricity $e = \frac{\sqrt{a^2-b^2}}{a}$, 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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Length of the curve is given by

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