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If the diameter of a circle is increased by 50%, what is the percentage increase

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Question: If the diameter of a circle is increased by 50%, what is the percentage increase in the area of the circle? (2021)

Options:

  1. 50%
  2. 75%
  3. 100%
  4. 125%

Correct Answer: 100%

Exam Year: 2021

Solution: 

If diameter increases by 50%, radius increases by 25%. Area increases by (new area - old area)/old area = (1.25² - 1) × 100% = 56.25%.

If the diameter of a circle is increased by 50%, what is the percentage increase

Practice Questions

Q1
If the diameter of a circle is increased by 50%, what is the percentage increase in the area of the circle? (2021)
  1. 50%
  2. 75%
  3. 100%
  4. 125%

Questions & Step-by-Step Solutions

If the diameter of a circle is increased by 50%, what is the percentage increase in the area of the circle? (2021)
  • Step 1: Understand that the diameter of a circle is the distance across the circle through its center.
  • Step 2: Know that the radius is half of the diameter.
  • Step 3: If the diameter increases by 50%, calculate the new diameter. For example, if the original diameter is D, the new diameter is D + 0.5D = 1.5D.
  • Step 4: Calculate the new radius. Since the radius is half of the diameter, the new radius is (1.5D) / 2 = 0.75D.
  • Step 5: The original radius is D / 2. To find the increase in radius, compare the new radius (0.75D) to the original radius (D / 2). The increase is (0.75D - D/2).
  • Step 6: Simplify the increase in radius. The original radius is D/2 = 0.5D, so the increase is 0.75D - 0.5D = 0.25D.
  • Step 7: Calculate the percentage increase in radius. The percentage increase is (increase/original) × 100% = (0.25D / 0.5D) × 100% = 50%.
  • Step 8: Now, calculate the area of the circle. The area A of a circle is given by the formula A = πr².
  • Step 9: Calculate the original area using the original radius (D/2): A_original = π(D/2)² = π(D²/4).
  • Step 10: Calculate the new area using the new radius (0.75D): A_new = π(0.75D)² = π(0.5625D²).
  • Step 11: Find the increase in area: Increase = A_new - A_original = π(0.5625D²) - π(D²/4).
  • Step 12: Simplify the increase in area: Increase = π(0.5625D² - 0.25D²) = π(0.3125D²).
  • Step 13: Calculate the percentage increase in area: Percentage increase = (Increase / A_original) × 100% = (π(0.3125D²) / π(D²/4)) × 100%.
  • Step 14: Simplify the percentage increase: Percentage increase = (0.3125D² / (D²/4)) × 100% = (0.3125 / 0.25) × 100% = 1.25 × 100% = 125%.
  • Step 15: Therefore, the percentage increase in the area of the circle is 56.25%.
  • Circle Geometry – Understanding the relationship between diameter, radius, and area of a circle.
  • Percentage Increase – Calculating the percentage increase based on changes in dimensions.
  • Area Formula – Using the formula for the area of a circle (A = πr²) to determine changes in area.
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