If the radius of a circle is increased by 50%, then what will be the percentage increase in the area of circle?

Options

A. 225
B. 125
C. 150
D. 175

Answer

To solve the problem of finding the percentage increase in the area of a circle when its radius is increased by 50%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Initial Radius**:
Let's denote the initial radius of the circle as \( r \).

2. **Calculate the Increased Radius**:
If the radius is increased by 50%, the new radius \( r' \) can be calculated as:
\[ r' = r + 0.5r = 1.5r \]

3. **Calculate the Initial Area**:
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Therefore, the initial area \( A \) is:
\[ A = \pi r^2 \]

4. **Calculate the New Area**:
Using the new radius \( r' \), the new area \( A' \) can be calculated as:
\[ A' = \pi (r')^2 = \pi (1.5r)^2 = \pi (2.25r^2) = 2.25\pi r^2 \]

5. **Calculate the Increase in Area**:
The increase in area \( \Delta A \) is given by:
\[ \Delta A = A' - A = 2.25\pi r^2 - \pi r^2 = (2.25 - 1)\pi r^2 = 1.25\pi r^2 \]

6. **Calculate the Percentage Increase**:
The percentage increase in area can be calculated using the formula:
\[ \text{Percentage Increase} = \left(\frac{\Delta A}{A}\right) \times 100 \]
Substituting the values we have:
\[ \text{Percentage Increase} = \left(\frac{1.25\pi r^2}{\pi r^2}\right) \times 100 = 1.25 \times 100 = 125\% \]

### Conclusion:
The percentage increase in the area of the circle when the radius is increased by 50% is **125%**.