Question

The ratio of the radii of two circles is 3:4, then the ratio of their areas is

• A. 3:4
• B. 4:3
• C. 9:16
• D. 16:9

Hint:

For any two similar 2D shapes, if the ratio of their corresponding lengths (like radius, side, etc.) is a:b, then the ratio of their areas will be \(a^2:b^2\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The area of a circle depends on the square of its radius. This question asks for the ratio of the areas of two circles given the ratio of their radii.

Step 2: Key Formula or Approach:
Area of a circle, \(A = \pi r^2\).
Let the radii of the two circles be \(r_1\) and \(r_2\), and their areas be \(A_1\) and \(A_2\).
The ratio of their areas is \(\frac{A_1}{A_2} = \frac{\pi r_1^2}{\pi r_2^2} = \left(\frac{r_1}{r_2}\right)^2\).

Step 3: Detailed Explanation:
We are given the ratio of the radii:
\[ \frac{r_1}{r_2} = \frac{3}{4} \] Now, we find the ratio of their areas using the formula derived in Step 2:
\[ \frac{A_1}{A_2} = \left(\frac{r_1}{r_2}\right)^2 = \left(\frac{3}{4}\right)^2 \] \[ \frac{A_1}{A_2} = \frac{3^2}{4^2} = \frac{9}{16} \] So, the ratio of the areas is 9:16.

Step 4: Final Answer:
The ratio of the areas of the two circles is 9:16. This matches option (C).