If the circumference of a circle increases from `2 pi" to 4 pi` then its area ......the original area :

Options

A. Half
B. Double
C. Three times
D. Four times

Answer

To solve the problem, we need to find out how the area of a circle changes when its circumference increases from \(2\pi\) to \(4\pi\).

### Step-by-Step Solution:

1. **Understand the Circumference Formula**:
The circumference \(C\) of a circle is given by the formula:
\[ C = 2\pi r \]
where \(r\) is the radius of the circle.

2. **Find the Original Radius**:
We know the original circumference is \(2\pi\). Setting this equal to the circumference formula:
\[ 2\pi = 2\pi r_1 \]
Dividing both sides by \(2\pi\):
\[ r_1 = 1 \]

3. **Calculate the Original Area**:
The area \(A\) of a circle is given by the formula:
\[ A = \pi r^2 \]
Substituting \(r_1 = 1\) into the area formula:
\[ A_1 = \pi (1^2) = \pi \]

4. **Find the New Radius**:
The new circumference is \(4\pi\). Setting this equal to the circumference formula:
\[ 4\pi = 2\pi r_2 \]
Dividing both sides by \(2\pi\):
\[ r_2 = 2 \]

5. **Calculate the New Area**:
Now, substituting \(r_2 = 2\) into the area formula:
\[ A_2 = \pi (2^2) = \pi \cdot 4 = 4\pi \]

6. **Compare the Areas**:
Now we compare the new area \(A_2\) with the original area \(A_1\):
\[ \text{Area increases from } A_1 = \pi \text{ to } A_2 = 4\pi \]
Thus, the area increases by a factor of:
\[ \frac{A_2}{A_1} = \frac{4\pi}{\pi} = 4 \]

### Conclusion:
The area increases to **4 times** the original area.