If the circumference of the circle is equal to twice its area, which of the following is equal to the area of this circle?

Options

A. `pi`
B. `2pi`
C. `4pi`
D. `16pi`

Answer

To solve the problem, we need to find the area of a circle given that its circumference is equal to twice its area. Let's break it down step by step.

### Step 1: Write down the formulas for circumference and area of a circle.
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]

### Step 2: Set up the equation based on the problem statement.
According to the problem, the circumference is equal to twice the area:
\[ C = 2A \]
Substituting the formulas for circumference and area, we get:
\[ 2\pi r = 2(\pi r^2) \]

### Step 3: Simplify the equation.
We can simplify the equation by dividing both sides by \( 2 \):
\[ \pi r = \pi r^2 \]
Next, we can divide both sides by \( \pi \) (assuming \( \pi \neq 0 \)):
\[ r = r^2 \]

### Step 4: Rearrange the equation.
Rearranging the equation gives:
\[ r^2 - r = 0 \]
Factoring out \( r \):
\[ r(r - 1) = 0 \]

### Step 5: Solve for \( r \).
Setting each factor to zero gives us:
\[ r = 0 \quad \text{or} \quad r = 1 \]
Since the radius of a circle cannot be zero, we have:
\[ r = 1 \]

### Step 6: Calculate the area of the circle.
Now that we have the radius, we can find the area:
\[ A = \pi r^2 = \pi (1)^2 = \pi \]

### Conclusion:
Thus, the area of the circle is:
\[ \boxed{\pi} \]