## The circumference of an ellipse

I was wondering about how to find the circumference of an ellipse.
I googled for it and found this: http://paulbourke.net/geometry/ellipsecirc/

That got me pretty amazed! I don't know high level maths, but still, can someone please explain to me why the circumference takes such a complicated form?

I mean, why shouldn't it simply be π*(a+b), where a & b are the major and minor axis.
 Recognitions: Homework Help http://www.mathsisfun.com/geometry/e...perimeter.html The best way to understand why the ellipse is so difficult to work out a formula for, try figuring out one for yourself. Apart from that it is difficult to figure what sort of answer you are expecting: it is the property of an ellipse to be like that just like it is the property of a circle to have an irrational ratio of circumference to diameter.

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 Quote by deep838 I was wondering about how to find the circumference of an ellipse. I googled for it and found this: http://paulbourke.net/geometry/ellipsecirc/ That got me pretty amazed! I don't know high level maths, but still, can someone please explain to me why the circumference takes such a complicated form? I mean, why shouldn't it simply be π*(a+b), where a & b are the major and minor axis.
What you are really asking is "why isn't everything trivial?". To which the only reasonable answer is "why should it be?"

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## The circumference of an ellipse

To get some intuition regarding why the circumference is not simply ##\pi(a+b)##, consider a highly eccentric ellipse, say with ##b >> a##. The circumference should not be very different from ##4b##, because the ellipse consists of two arcs from ##(0,b)## to ##(0,-b)## (assuming appropriately chosen coordinates) which are nearly straight line segments, each of length ##2b##. Thus the circumference should be close to ##4b##, whereas your proposed formula gives ##\pi(a+b) \approx \pi b##.

Therefore, for ##b >> a##, the ##\pi(a+b)## formula would need to be multiplied by a correction factor of approximately ##4/\pi \approx 1.27##.

Compare this with the "better" approximation given here, for example: http://en.wikipedia.org/wiki/Ellipse#Area

$$\pi(a+b) \left(1 + \frac{3\left(\frac{a-b}{a+b}\right)^2}{10 + \sqrt{4 - 3\left(\frac{a-b}{a+b}\right)^2}}\right)$$
We may view the expression in the large parentheses as a correction factor applied to ##\pi(a+b)##. If ##b>>a## we may approximate ##a \approx 0## in that expression, and the result is
$$1 + \frac{3}{10 + \sqrt{4 - 3}} = 1 + \frac{3}{11} \approx 1.27$$
 hmm... thanks to all of you... especially jbuniii !!! i can now see what i wasn't seeing before! and forgive me if i'm asking too much, but can any of you provide me a link to where this/these expressions are derived?

 Tags circumference, ellipse, precision