If the circumference of a circle is increased by 50% then the area will be increased by

Options

A. (a) `50%`
B. (b) `75%`
C. (c) `100%`
D. (d) `125%`

Answer

To solve the problem of how much the area of a circle increases when its circumference is increased by 50%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between circumference and radius**:
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi R \]
where \( R \) is the radius of the circle.

2. **Calculate the new circumference**:
If the circumference is increased by 50%, the new circumference \( C' \) can be calculated as:
\[ C' = C + 0.5C = 1.5C = 1.5 \times 2\pi R = 3\pi R \]

3. **Find the new radius**:
The new circumference \( C' \) can also be expressed in terms of the new radius \( R' \):
\[ C' = 2\pi R' \]
Setting the two expressions for \( C' \) equal gives:
\[ 3\pi R = 2\pi R' \]
Dividing both sides by \( 2\pi \):
\[ R' = \frac{3R}{2} \]

4. **Calculate the old area**:
The area \( A \) of the original circle is given by:
\[ A = \pi R^2 \]

5. **Calculate the new area**:
The area \( A' \) of the new circle with radius \( R' \) is:
\[ A' = \pi (R')^2 = \pi \left(\frac{3R}{2}\right)^2 = \pi \left(\frac{9R^2}{4}\right) = \frac{9\pi R^2}{4} \]

6. **Find the increase in area**:
The increase in area \( \Delta A \) is:
\[ \Delta A = A' - A = \frac{9\pi R^2}{4} - \pi R^2 = \frac{9\pi R^2}{4} - \frac{4\pi R^2}{4} = \frac{5\pi R^2}{4} \]

7. **Calculate the percentage increase in area**:
The percentage increase in area can be calculated as:
\[ \text{Percentage Increase} = \left(\frac{\Delta A}{A}\right) \times 100 = \left(\frac{\frac{5\pi R^2}{4}}{\pi R^2}\right) \times 100 = \left(\frac{5}{4}\right) \times 100 = 125\% \]

### Conclusion:
Thus, if the circumference of a circle is increased by 50%, then the area will be increased by **125%**.