Question

The circumferences of two circles are in the ratio 2 : 3. The ratio between their areas is
(a) 2 : 3
(b) 4 : 9
(c) 9 : 4
(d) none of these

Answer 1

(b) 4:9
Let $r_1\mathrm{cm}\mathrm{and}{r}_2$ cm be the radii of the two circles.
Thus, we have:
Perimeter of the first circle$=2\mathrm{\pi }{r}_1$
And,
Perimeter of the second circle$=2\mathrm{\pi }{r}_2$
Now,
$$\begin{gathered} \frac{2\mathrm{\pi }{r}_1}{2\mathrm{\pi }{r}_2}=\frac{2}{3} \\ \Rightarrow \frac{r_1}{r_2}=\frac{2}{3} \end{gathered}$$
Also,
Area of the first circle$=\mathrm{\pi }{r_1}^2$
And,
Area of the second circle$=\mathrm{\pi }{r_2}^2$
Thus, we have:
$\frac{\mathrm{\pi }{r_1}^2}{\mathrm{\pi }{r_2}^2}=\frac{{r_1}^2}{{r_2}^2}$

$$\begin{gathered} =\frac{2^2}{3^2}\left[\because \frac{{\mathrm{r}}_1}{{\mathrm{r}}_2}=\frac{2}{3}\right] \\ =\frac{4}{9} \\ =4:9 \end{gathered}$$

Hence, the ratio of the areas of the two circles is 4:9.