Question

The radius of a circle is increased by 50%. What is the percent increase in its area?

• A. 50%
• B. 125%
• C. 75%
• D. 100%

Answer 1

Answer:

B. 125%

Understanding Percentage Increase in Circle Area

Key Concepts:

• The area of a circle is calculated using the formula: $A = \pi r^2$, where $A$ is the area and $r$ is the radius.

• A percentage increase means finding out how much a quantity has grown relative to its original value, expressed as a fraction of 100.

Problem Breakdown:

• When the radius of a circle is increased by 50%, the new radius becomes 1.5 times the original radius.

• Let the original radius be $r$. The new radius will be $r + 0.50r = 1.50r$.

• The original area of the circle is $A_{original} = \pi r^2$.

• The new area of the circle is $A_{new} = \pi (1.50r)^2 = \pi (2.25r^2) = 2.25 \pi r^2$.

• The increase in area is $A_{new} - A_{original} = 2.25 \pi r^2 - \pi r^2 = 1.25 \pi r^2$.

• To find the percent increase in area, we use the formula: $\left( \frac{\text{Increase in Area}}{\text{Original Area}} \right) \times 100\%$.

• Percent Increase $= \left( \frac{1.25 \pi r^2}{\pi r^2} \right) \times 100\% = 1.25 \times 100\% = 125\%$.