Question

The circumferences of two circles are in the ratio 3 : 4. The ratio of their areas is
(a) 3: 4
(b) 4 : 3
(c) 9 : 16
(d) 16: 9

Answer 1

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,
$$\begin{gathered} \frac{c}{C}=\frac{3}{4} \\ \Rightarrow \frac{2\mathrm{\pi }r}{2\mathrm{\pi }R}=\frac{3}{4} \\ \Rightarrow \frac{r}{R}=\frac{3}{4} \end{gathered}$$
Now, the ratio between their areas is given by
$$\begin{gathered} \frac{a}{A}=\frac{\mathrm{\pi }{r}^2}{\mathrm{\pi }{R}^2} \\ ={\left(\frac{r}{R}\right)}^2 \\ ={\left(\frac{3}{4}\right)}^2 \\ =\frac{9}{16} \end{gathered}$$
Hence, the correct answer is option (c).