WLOG we can assume $a<b$. Therefore, if $\frac{b}{a} \ge \frac{2 \sqrt{3}}{3}$, the answer will be $\frac{2 \sqrt{3}}{3}a$; if $\frac{b}{a} \lt \frac{2 \sqrt{3}}{3}$, the answer will be $2 \sqrt{a^{2}+b^{2}-\sqrt{3}ab}$.