To find the area of a circle given its circumference \( C \), we can use the formulas for circumference and area of a circle.
The formula for the circumference is: \[ C = 2\pi r \] where \( r \) is the radius of the circle.
We are given that: \[ C = 4\pi \]
Now, we can set the two expressions for circumference equal to each other and solve for \( r \): \[ 2\pi r = 4\pi \]
Dividing both sides by \( 2\pi \): \[ r = \frac{4\pi}{2\pi} = 2 \]
Now that we have the radius, we can find the area \( A \) of the circle using the area formula: \[ A = \pi r^2 \]
Substituting \( r = 2 \): \[ A = \pi (2^2) = \pi \times 4 = 4\pi \]
Thus, the area of the circle is: \[ \boxed{4\pi} \]