**Question:**

The circumferences of two circles are in the ratio 2: 3. What is the ratio between their areas?

**Solution:**

Let the 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,

$\frac{c}{C}=\frac{2}{3}$

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

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

Now, the ratio between their areas is given by

$\frac{a}{A}=\frac{\pi r^{2}}{\pi R^{2}}$

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

$=\left(\frac{2}{3}\right)^{2}$

$=\frac{4}{9}$

Hence, the ratio between their areas is 4 : 9.