Same Arc Length

If an arc of $45^\circ$ on circle $A$ has the same length as an arc of $30^\circ$ on circle $B$, find the ratio of the area of circle $A$ to the area of circle $B$.
Source: NCTM Mathematics Teacher

SOLUTION

Let $r_A$ be the radius of circle $A$ and $r_B$ the radius of circle $B$.
$circunference\;of\;A=2\pi r_A$
$circumference\;of\;A=\displaystyle\frac{360^\circ}{45^\circ}\left (a\right )=8a$
$2\pi r_A=8a$

$circumferencce\;of\;B=2\pi r_B$
$circumference\;of\;B=\displaystyle\frac{360^\circ}{30^\circ}\left (a\right )=12a$
$2\pi r_B=12a$

$\displaystyle\frac{2\pi r_A}{2\pi r_B}=\frac{8a}{12a}$
$\displaystyle\frac{r_A}{r_B}=\frac{2}{3}$

$\displaystyle\frac{area\;of\;A}{area\;of\;B}=\frac{\pi r_A^2}{\pi r_B^2}$
$=\displaystyle\frac{r_A^2}{r_B^2}$
$=\left (\displaystyle\frac{r_A}{r_B}\right )^2$
$=\left (\displaystyle\frac{2}{3}\right )^2$
$=\displaystyle\frac{4}{9}$

Answer: $\displaystyle\frac{4}{9}$