smc

Let $r_1$ and $r_2$ be the radii of circles $A$ and$B$, respectively. It is well known that in a circle with radius $r$, a subtended arc opposite an angle of $\theta$ degrees has length $\frac{\theta }{360}\cdot 2\pi r$. Using that here, the arc of circle A has length $\frac{45}{360}\cdot 2\pi r_1=\frac{r_1\pi }{4}$. The arc of circle B has length $\frac{30}{360}\cdot 2\pi r_2=\frac{r_2\pi }{6}$. We know that they are equal, so $\frac{r_1\pi }{4}=\frac{r_2\pi }{6}$, so we multiply through and simplify to get $\frac{r_1}{r_2}=\frac{2}{3}$. As all circles are similar to one another, the ratio of the areas is just the square of the ratios of the radii, so our answer is $\boxed{4/9}$.