If the radius of a circle is reduced by 50%, by what percentage is the area reduced?

Introduction / Context:The area of a circle depends on the square of the radius. Therefore, any percentage change in the radius translates to a squared effect on area. Halving the radius is a common benchmark scenario that sharply reduces the area, and recognizing the squared relationship is key to avoiding arithmetic traps.

Given Data / Assumptions:

• Original radius r.
• New radius r′ = 0.5r (i.e., reduced by 50%).
• Area formula: A = πr^2.

Concept / Approach:Apply the scale factor on radius to the area. If radius is multiplied by k, area is multiplied by k^2. Here k = 0.5, hence area factor is (0.5)^2 = 0.25. The reduction is thus 1 − 0.25 = 0.75 = 75%.

Step-by-Step Solution:Original area A = πr^2New area A′ = π(0.5r)^2 = π * 0.25 r^2 = 0.25APercentage decrease = (A − A′) / A * 100% = (1 − 0.25) * 100% = 75%

Verification / Alternative check:Use numbers: let r = 10 ⇒ A = 100π. Halve radius to 5 ⇒ A′ = 25π. Decrease = 75π, which is 75% of 100π — consistent.

Why Other Options Are Wrong:65.5%, 39.5%, and 34.5% come from unrelated combinations or linear thinking. Area does not change linearly with radius; it changes with the square of the radius scale factor.

Common Pitfalls:Adding or subtracting percentages linearly, or assuming 50% less radius gives 50% less area. The squared dependence makes the effect much larger: 75% reduction.

Final Answer:75%