Question

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

Answer 1

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,
$$\begin{gathered} \frac{a}{A}=\frac{4}{9} \\ \Rightarrow \frac{\mathrm{\pi }{r}^2}{\mathrm{\pi }{R}^2}={\left(\frac{2}{3}\right)}^2 \\ \Rightarrow \frac{r}{R}=\frac{2}{3} \end{gathered}$$
Now, the ratio between their circumferences is given by
$$\begin{gathered} \frac{c}{C}=\frac{2\mathrm{\pi }r}{2\mathrm{\pi }R} \\ =\frac{r}{R} \\ =\frac{2}{3} \end{gathered}$$
Hence, the ratio between their circumferences is 2 : 3.