If the radius of a circle is decreased by 50%, what is the percentage decrease in the area of the circle (report the decrease, not the remaining area)?

Introduction:
This question checks understanding of how area scales with radius in a circle. The area of a circle is proportional to the square of its radius. Therefore, reducing the radius by a certain fraction causes a much larger reduction in area because the radius is squared. Many mistakes happen when learners assume area decreases by the same percentage as radius. The correct method is to convert the radius change into a scale factor and then square it to find the new area factor.

Given Data / Assumptions:

• Radius is decreased by 50%• Original radius = r• Circle area formula: A = pi * r^2

Concept / Approach:
If radius decreases by 50%, new radius = r * (1 - 50/100) = 0.5r. Since area depends on r^2, new area factor = (0.5)^2 = 0.25. That means the circle retains 25% of its original area, so the decrease is 75%.

Step-by-Step Solution:
Step 1: Let original radius = r.Original area = pi * r^2Step 2: Decrease radius by 50%.New radius = 0.5rStep 3: Compute new area.New area = pi * (0.5r)^2 = pi * (0.25r^2) = 0.25 * (pi * r^2)So new area = 25% of original areaStep 4: Convert to percentage decrease.Decrease = 100% - 25% = 75%

Verification / Alternative check:
Take r = 10. Original area = pi*100. New radius = 5, new area = pi*25. New/original = 25/100 = 0.25 (25%). Decrease = 75%. This confirms the squared scaling relationship is applied correctly.

Why Other Options Are Wrong:
50% is the common incorrect assumption that area changes linearly with radius.55%, 65%, 85% are random deviations and do not match the exact (0.5)^2 scaling.Only 75% correctly reflects that the area becomes one-fourth of the original.

Common Pitfalls:
• Applying the percentage change to area directly instead of to radius.• Forgetting that area uses r^2.• Confusing “decrease in area” with “new area percentage”.

Final Answer:
The area decreases by 75%.