If the radius of a circle is increased by 25%, by what percentage does the area of the circle increase?

Introduction / Context:
This is a classic percentage change question in geometry that examines your understanding of how the area of a circle depends on its radius. The radius is increased by a certain percentage, and you must determine the resulting percentage increase in the area. This kind of problem emphasizes that when a quantity depends on the square of a variable, small changes in the variable can produce larger changes in the dependent quantity.

Given Data / Assumptions:

• Original radius of the circle is r.
• The radius is increased by 25% to become the new radius.
• We must find the percentage increase in the area of the circle.
• The relationship between area and radius for a circle is used in its standard form.

Concept / Approach:
The area of a circle is given by:
A = π * r^2. If the radius is increased by 25%, the new radius r_new is:
r_new = r * (1 + 25 / 100) = r * 1.25. The new area A_new becomes π * r_new^2. We compare A_new with the original area A and compute the percentage increase as:
Percentage increase = [(A_new - A) / A] * 100 percent.

Step-by-Step Solution:
Step 1: Original area: A = π * r^2. Step 2: New radius: r_new = 1.25 * r. Step 3: New area: A_new = π * (r_new)^2 = π * (1.25r)^2. Step 4: Compute (1.25)^2: 1.25 * 1.25 = 1.5625. Step 5: So A_new = π * 1.5625 * r^2 = 1.5625 * A. Step 6: Increase in area = A_new - A = (1.5625A - A) = 0.5625A. Step 7: Percentage increase = (0.5625A / A) * 100 percent = 0.5625 * 100 percent = 56.25 percent. Step 8: So the area increases by 56.25 percent.

Verification / Alternative check:
For a quick check, assume a convenient radius. Let r = 4 units. Original area A = π * 4^2 = 16π. New radius is r_new = 1.25 * 4 = 5. New area A_new = π * 5^2 = 25π. The increase is 25π - 16π = 9π. Percentage increase is (9π / 16π) * 100 percent = (9 / 16) * 100 percent = 56.25 percent. This numerical example confirms the algebraic result.

Why Other Options Are Wrong:
An increase of 25 percent would be correct if the area depended linearly on the radius, but it depends on the square of the radius. Options 50 percent and 28.125 percent do not match the correct squared factor of 1.5625. The value 31.25 percent would correspond to a much smaller change and does not come from squaring a 25 percent increase in radius. Only 56.25 percent correctly reflects the transformed area.

Common Pitfalls:
A very common mistake is to assume that if the radius increases by 25 percent, the area also increases by 25 percent. This ignores the square relationship between area and radius. Another error is squaring 25 instead of 1.25, leading to 625 percent or other meaningless values. Some students also forget to subtract 1 when converting the factor 1.5625 into a percentage increase. Careful stepwise reasoning avoids these errors.

Final Answer:
Thus, when the radius is increased by 25%, the area of the circle increases by 56.25 percent.