If the circumfrence of a circle increasses from `4pi` to `8pi`, then find the percentage increase in the area of the circle.

Answer

To find the percentage increase in the area of a circle when its circumference increases from \(4\pi\) to \(8\pi\), we can follow these steps:

### Step 1: Calculate the radius from the initial circumference
The formula for the circumference \(C\) of a circle is given by:
\[ C = 2\pi r \]
where \(r\) is the radius.

For the initial circumference \(C_1 = 4\pi\):
\[ 4\pi = 2\pi r_1 \]
Dividing both sides by \(2\pi\):
\[ r_1 = \frac{4\pi}{2\pi} = 2 \]

### Step 2: Calculate the area from the initial radius
The area \(A\) of a circle is given by:
\[ A = \pi r^2 \]
For the initial radius \(r_1 = 2\):
\[ A_1 = \pi (2)^2 = \pi \cdot 4 = 4\pi \]

### Step 3: Calculate the radius from the final circumference
Now, for the final circumference \(C_2 = 8\pi\):
\[ 8\pi = 2\pi r_2 \]
Dividing both sides by \(2\pi\):
\[ r_2 = \frac{8\pi}{2\pi} = 4 \]

### Step 4: Calculate the area from the final radius
For the final radius \(r_2 = 4\):
\[ A_2 = \pi (4)^2 = \pi \cdot 16 = 16\pi \]

### Step 5: Calculate the increase in area
The increase in area \(\Delta A\) is given by:
\[ \Delta A = A_2 - A_1 = 16\pi - 4\pi = 12\pi \]

### Step 6: Calculate the percentage increase in area
The percentage increase in area is calculated using the formula:
\[ \text{Percentage Increase} = \left(\frac{\Delta A}{A_1}\right) \times 100 \]
Substituting the values:
\[ \text{Percentage Increase} = \left(\frac{12\pi}{4\pi}\right) \times 100 = 3 \times 100 = 300\% \]

### Final Answer
The percentage increase in the area of the circle is \(300\%\).

---