The areas of two circles are in the ratio 4: 9.

Question:

The areas of two circles are in the ratio 4: 9. What is the ratio between their circumferences?

Solution:

Let the radii of the two circles be r and R, the circumferences of the circles be c and C and the areas of the two circles be a and A.
Now,

$\frac{a}{A}=\frac{4}{9}$

$\Rightarrow \frac{\pi r^{2}}{\pi R^{2}}=\left(\frac{2}{3}\right)^{2}$

$\Rightarrow \frac{r}{R}=\frac{2}{3}$

Now, the ratio between their circumferences is given by

$\frac{c}{C}=\frac{2 \pi r}{2 \pi R}$

$=\frac{r}{R}$

$=\frac{2}{3}$

Hence, the ratio between their circumferences is 2 : 3.