# 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.

Last edited:

Simon Bridge
Homework Helper
http://www.mathsisfun.com/geometry/ellipse-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.

HallsofIvy
Homework Helper
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?"

jbunniii
Homework Helper
Gold Member
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?