Let $Q$ be an ellipse with integer-length axes $a$ and $b$: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \;.$$ The perimeter of $Q$ is given by the complete elliptic integral of the 2nd kind, $E(\;)$: $4 a E(\epsilon)$ where the eccentricity $\epsilon = \sqrt{1-(b/a)^2}$. Is the following known?

Q. Is the perimeter of an ellipse with integer-length axes $a$ and $b$ always irrational? If not, for which $a,b$ is it rational?

_{$a,b=2,1$. Perimeter is approximately $9.051752335744220113037500073330267105760842885887$.}

**Answered**by

*paul Monsky*in the comments: The perimeter is transcendental! Established by Theodor Schneider in 1937.