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

Let the radii of two circles are \( \Large r_{1} \) and \( \Large r_{2} \) respectively.

Given,

\( \Large \frac{Circumference \ of \ 1st \ circle}{Circumference \ of \ 2nd \ circle}=\frac{2}{3} \)

=> \( \Large \frac{2\pi r_{1}}{2\pi r_{2}}=\frac{2}{3} \)=> \( \Large \frac{r_{1}}{r_{2}}=\frac{2}{3} \) => \( \Large \left(\frac{r_{1}}{r_{2}}\right)^{2}=\frac{4}{9} \)

\( \Large \frac{Area \ of \ 1st \ circle}{Area \ of \ 2nd \ circle} \)= \( \Large \left(\frac{r_{1}}{r_{2}}\right)^{2}=\frac{4}{9} \)