If the radius of a circle is increased by `6 %` , find the percentage increase in its area

Options

A. `15 %`
B. `12.36 %`
C. `8.39 %`
D. `17 %`

Answer

To find the percentage increase in the area of a circle when the radius is increased by 6%, we can follow these steps:

### Step 1: Understand the relationship between radius and area
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.

### Step 2: Define the original radius
Let the original radius of the circle be \( r \).

### Step 3: Calculate the new radius
If the radius is increased by 6%, the new radius \( r' \) can be calculated as:
\[ r' = r + 0.06r = 1.06r \]

### Step 4: Calculate the original area
Using the original radius, the original area \( A \) is:
\[ A = \pi r^2 \]

### Step 5: Calculate the new area
Using the new radius \( r' \), the new area \( A' \) is:
\[ A' = \pi (r')^2 = \pi (1.06r)^2 \]
\[ A' = \pi (1.1236r^2) = 1.1236\pi r^2 \]

### Step 6: Find the percentage increase in area
The percentage increase in area can be calculated using the formula:
\[ \text{Percentage Increase} = \left(\frac{A' - A}{A}\right) \times 100 \]
Substituting the values of \( A' \) and \( A \):
\[ \text{Percentage Increase} = \left(\frac{1.1236\pi r^2 - \pi r^2}{\pi r^2}\right) \times 100 \]
\[ = \left(\frac{1.1236 - 1}{1}\right) \times 100 \]
\[ = 0.1236 \times 100 = 12.36\% \]

### Conclusion
The percentage increase in the area of the circle when the radius is increased by 6% is approximately **12.36%**.

---