Circle scaling — If the radius of a circle is increased by 6%, find the percentage increase in its area.

Introduction / Context:
This is a percentage change under geometric scaling. Area of a circle is proportional to r^2, so a change in radius scales the area by the square of the same linear factor.

Given Data / Assumptions:

• Radius increases by 6% ⇒ new radius factor = 1.06
• Area ∝ r^2

Concept / Approach:
If r → 1.06r, then area A → (1.06)^2 * A. The percentage increase is (1.06^2 − 1) * 100%.

Step-by-Step Solution:
1.06^2 = 1.1236Increase = (1.1236 − 1) * 100% = 0.1236 * 100% = 12.36%

Verification / Alternative check:
Use small-change approximation: for small x, (1 + x)^2 ≈ 1 + 2x = 1 + 12%. Exact result adds x^2 = 0.36%, giving 12.36%.

Why Other Options Are Wrong:
10.25% corresponds to a 5% linear increase, not 6%; 15% and 17% are overestimates; 8.39% is unrelated.

Common Pitfalls:
Doubling the 6% directly (i.e., 12%) without the square term; here, the extra 0.36% comes from 0.06^2.

Final Answer:
12.36%