Effect of Doubling Diameter on Circle Area — Percent Increase: If the diameter of a circle is increased by 100% (i.e., doubled), by what percent does its area increase?
Aptitude
Area
Difficulty: Easy
Choose an option
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A100%
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B200%
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C300%
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D400%
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E150%
Answer
Correct Answer: 300%
Explanation
Introduction / Context:Circle area depends on the square of a linear dimension (radius or diameter). Doubling the diameter doubles the radius as well, so the area multiplies by 2^2 = 4. Converting that factor to a percentage increase distinguishes absolute area from relative change.
Given Data / Assumptions:
- Original diameter D; new diameter = 2D
- Original radius r = D/2; new radius r' = D
- Area formula: A = πr^2
Concept / Approach:Compute the factor by which area changes: A'/A = (r'^2)/(r^2) = (D^2)/(D^2/4) = 4. A factor of 4 corresponds to a 300% increase because Increase% = (new − old)/old * 100% = (4A − A)/A * 100% = 300%.
Step-by-Step Solution:
Original area A = π(D/2)^2 = πD^2/4.New area A' = π(D)^2 = πD^2.Ratio A'/A = (πD^2) / (πD^2/4) = 4 ⇒ Increase% = (4 − 1)*100% = 300%.Verification / Alternative check:
Numeric example: Let D = 10 ⇒ A = 78.54. New D = 20 ⇒ A' = 314.16. Increase = 235.62 ≈ 300% of 78.54.Why Other Options Are Wrong:
- 100% implies doubling area, not quadrupling.
- 200% implies tripling total, still short of 4A.
- 400% confuses new area with the percent increase.
- 150% is arbitrary and unsupported.
Common Pitfalls:
- Reporting the new area as 400% instead of the increase of 300%.
Final Answer:300% increase.