Perimeter of an ellipse

For general closed curve(preferably loop), perimeter=$\int_0^{2\pi}rd\theta$ where (r,$\theta$) represents polar coordinates.

In ellipse, $r=\sqrt {a^2\cos^2\theta+b^2\sin^2\theta}$

So, perimeter of ellipse = $\int_0^{2\pi}\sqrt {a^2\cos^2\theta+b^2\sin^2\theta}d\theta$

I don't know if closed form for the above integral exists or not, but even if it doesn't have a closed form , you can use numerical methods to compute this definite integral.

Generally, people use an approximate formula for arc length of ellipse = $2\pi\sqrt{\frac{a^2+b^2}{2}}$

you can also visit this link : http://pages.pacificcoast.net/~cazelais/250a/ellipse-length.pdf