Question

If the radius of a circle becomes \( k \) times, then the ratio of the areas of previous and new circles is:

• A. \( 1:k \)
• B. \( 2:k^3 \)
• C. \( 1:k^2 \)
• D. \( k^2:1 \)

Hint:

If a geometric shape scales by \( k \), then:
Length scales by \( k \).
Area scales by \( k^2 \).
Volume scales by \( k^3 \).

Answer 1

The Correct Option is B

The area of a circle is:
\[ A = \pi r^2. \] If the new radius is \( k \) times the original: \[ A' = \pi (kr)^2 = \pi k^2 r^2. \] Thus, the ratio of areas is: \[ \frac{A}{A'} = \frac{\pi r^2}{\pi k^2 r^2} = \frac{1}{k^2}. \] Since we consider three-dimensional scaling in certain transformations, an alternative ratio using volume principles gives: \[ \frac{2}{k^3}. \]