Area ratio from radius ratio: The radii of two circles are in the ratio 1 : 3. What is the ratio of their areas?

Introduction / Context:
For similar figures like circles, linear measures scale by k, while areas scale by k^2. Translating a radius ratio into an area ratio is a fundamental proportional reasoning task.

Given Data / Assumptions:

• r1 : r2 = 1 : 3
• Area of a circle A = πr^2

Concept / Approach:
If r2 = 3 * r1, then A2/A1 = (π * (3r1)^2) / (π * r1^2) = 9. Hence A1 : A2 = 1 : 9. Always square the scale factor for areas.

Step-by-Step Solution:
Let r1 = 1k, r2 = 3kA1 = π * (1k)^2 = πk^2A2 = π * (3k)^2 = 9πk^2A1 : A2 = 1 : 9

Verification / Alternative check:
Pick numbers: r1 = 1, r2 = 3 → A1 = π, A2 = 9π → ratio 1:9, independent of π or k.

Why Other Options Are Wrong:
1:3 confuses linear with area scaling; 1:6 and 3:9 are inconsistent; “None” is incorrect since 1:9 is definitive.

Common Pitfalls:
Not squaring the ratio for areas; mixing up which circle is first in the ratio.

Final Answer:
1:9