The ratio of the circumference of two circles is 7:11. What is the ratio of the areas of the two circles?

Let r₁ and r₂ be the radius of the two circles.

It is known that the circumference of a circle is given by 2πr.

∴ Circumference of circle with radius r₁ = 2πr₁

Circumference of circle with radius r₂ = 2πr₂

It is given that the ratio of circumference of two circles is 7: 11.

$$\begin{gathered} \therefore \frac{2\mathrm{\pi }{r}_1}{2\mathrm{\pi }{r}_2}=\frac{7}{11} \\ \Rightarrow \frac{r_1}{r_2}=\frac{7}{11} \end{gathered}$$

It is known that the area of a circle is given by πr².

∴ Ratio of areas of the circles

 $=\frac{\mathrm{\pi }{r_1}^2}{\mathrm{\pi }{r_2}^2}=\frac{{r_1}^2}{{r_2}^2}={\left(\frac{r_1}{r_2}\right)}^2={\left(\frac{7}{11}\right)}^2=\frac{49}{121}$

Thus, the ratio of the areas of the two circles is 49: 121.