Question

If the circumferences of two circles are in the ratio c₁ : c₂, then the ratio of their areas is __________.

Answer 1

Let R and r be the radii of the two circles.

Now,

Circumference of first circle : Circumference of second circle = c₁ : c₂ (Given)

$$\begin{gathered} \therefore \frac{2\mathrm{\pi }R}{2\mathrm{\pi }r}=\frac{c_1}{c_2} \\ \Rightarrow \frac{R}{r}=\frac{c_1}{c_2} \end{gathered}$$

Squaring on both sides, we get

$$\begin{gathered} \frac{R^2}{r^2}=\frac{c_1^2}{c_2^2} \\ \Rightarrow \frac{\mathrm{\pi }{R}^2}{\mathrm{\pi }{r}^2}=\frac{c_1^2}{c_2^2} \\ \Rightarrow \frac{\mathrm{Area}\mathrm{of}\mathrm{first}\mathrm{circle}}{\mathrm{Area}\mathrm{of}\mathrm{second}\mathrm{circle}}=\frac{c_1^2}{c_2^2} \end{gathered}$$

∴ Area of first circle : Area of second circle = $c_1^2:c_2^2$

If the circumferences of two circles are in the ratio c₁ : c₂, then the ratio of their areas is $\underline{c_1^2:c_2^2}$.