### Circumference

Let $x=a\cos t$, $y=b\sin t$, $0\leq t <2\pi$. Then
$ds^2=dx^2 + dy^2=\left[(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2\right] dt^2=(a^2\sin^2 t+ b^2\cos^2 t)dt^2\\ s=\int ds=\int_0^{2\pi} \sqrt{a^2\sin^2 t+ b^2\cos^2 t} dt$