If the radius of a circle is increased by 15%, by what percentage does the area of the circle increase?

Introduction / Context:

This question links percentage change in a geometric measurement (radius) to the resulting change in area. It tests your understanding that area of a circle depends on the square of the radius. A percentage increase in radius does not produce the same percentage increase in area; instead, the effect is amplified due to the square relationship.

Given Data / Assumptions:

• The radius of a circle is increased by 15%.
• The original radius is r and the new radius is 1.15 * r.
• Area of a circle is proportional to the square of its radius.
• We must find the percentage increase in area.

Concept / Approach:

The area of a circle is A = pi * r^2. If the radius changes to r_new, the new area becomes pi * r_new^2. The ratio of new area to old area is (r_new^2) / (r^2). When the radius is multiplied by 1.15, the area is multiplied by (1.15)^2. The percentage increase in area is then (area factor − 1) * 100.

Step-by-Step Solution:

Verification / Alternative check:

To verify, take an example. Suppose the original radius is 10 units. Then original area is proportional to 10^2 = 100 units squared. New radius is 11.5 units, and area is proportional to 11.5^2 = 132.25 units squared. Increase in area = 132.25 − 100 = 32.25 units squared. Percentage increase = 32.25 / 100 * 100 = 32.25%. This confirms the calculation.

Why Other Options Are Wrong:

• 30%: This is close but ignores the exact effect of squaring 1.15.
• 16.125%: This is roughly half of the correct value and might come from incorrectly applying the percentage to only one factor.
• 25% and 15%: These treat the change in radius and area as if they are directly proportional, which is not correct because area depends on the square of the radius.

Common Pitfalls:

A typical mistake is to assume that the percentage change in area equals the percentage change in radius. Another error is to compute 15% of the area and add it once, ignoring the square effect. Always remember that when a dimension is squared in a formula, percentage changes must be handled by applying the change factor and then squaring it, not by squaring the percentage itself.

Final Answer:

The area of the circle increases by 32.25%.