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.
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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 quasiperiod 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 axeslengths is transcendental. 

