Is there any ellipse with nonzero rational minor and major axis lengths $a$ and $b$ such that the circumference of the ellipse is rational too? (or the weaker variant: ... such that the circumference is algebraic?)  

Note that the surface area $S=\pi ab$ is of course always transcendental in such a situation.



any ellipse with rational axes and circumference?
No, there is no such ellipse. This is exactly theorem $6.5$ of Alan Baker's book TRANSCENDENTAL
NUMBER THEORY, as pointed out by Felipe Voloch.


Let $\omega$ be a primitive period of a $\wp$-function with algebraic
invariants $g\_2, g\_3$ and let $\eta=2\zeta(\omega /2)$ be the associated quasi-period of the
Weierstrass $\zeta$-function satisfying $\zeta'(z) = -\wp (z)$. Then we have


**Theorem $6.5$.** Any linear combination of $\omega$ and $\eta$ with algebraic
coefficients, not both zero, is transcendental.


Because $\omega$ and $\eta$ can be represented as elliptic integrals of the first and second
kinds respectively, Theorem 6.5 implies that the circumference of any ellipse with algebraic axes-lengths is transcendental.


