The ratio of the circumferences of two circles is 2 : 3. Find the ratio of their areas.
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 4 : 9
The circumference \(2\pi r\) is proportional to the radius, while the area \(\pi r^{2}\) is proportional to the square of the radius. So the ratio of the circumferences is the ratio of the radii, and squaring it gives the ratio of the areas.
Ratio of the circumferences, which is the ratio of the radii: \(\displaystyle\frac{2\pi r_1}{2\pi r_2}=\frac{r_1}{r_2}=\frac{2}{3}\)
Ratio of the areas, the square of that ratio: \(\displaystyle\frac{\pi r_1^{2}}{\pi r_2^{2}}=\left(\frac{2}{3}\right)^{2}=\frac{4}{9}\)