Circumference length of an ellipse

Question

I tried to find out the circumference of an ellipse many times by using integration but got stuck in the middle. So I tried to search on Google. Some sites said that it is bit difficult to find the circumference of ellipse and gave me some approximations which produce almost accurate results. But none of them gave me the original formula. Is there any such formula where I can just put the values of minor and major axis to get its accurate circumference?

Answer 1

The circumference of the ellipse (or the length of an arc of ellipse) can't be expressed with the elementary functions: it depends on the elliptic integral of the second kind: $$E(k)=\int_0^{\tfrac\pi2}\sqrt{1-k^2\sin^2t\mathstrut}\,\mathrm dt. $$ If the ellipse has semi-major and semi-minor axes $a$ and $b$ respectively, the circumference is equal to $$4aE(e),\qquad \text{ where }\enspace e=\sqrt{1-\dfrac{b^2}{a^2}}\enspace\text{(eccentricity)}.$$ You only can have an expansion as a power series for the function $E(k)$: $$E(k)=\frac\pi2\biggl(1-\Bigl(\frac 12\Bigr)^{\!2}\frac{k^2}1 -\Bigl(\frac {1\cdot 3}{2\cdot 4}\Bigr)^{\!2}\frac{k^4}3-\Bigl(\frac {1\cdot 3\cdot5}{2\cdot 4\cdot6}\Bigr)^{\!2}\frac{k^6}5-\dotsm$$