The formula for the circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle.
Given that the circumference \( C = 4\pi \), we can set up the equation:
\[ 2\pi r = 4\pi \]
Now, divide both sides by \( 2\pi \):
\[ r = \frac{4\pi}{2\pi} = 2 \]
Now that we have the radius \( r = 2 \), we can find the area \( A \) of the circle using the formula:
\[ A = \pi r^2 \]
Substituting the radius into the area formula gives:
\[ A = \pi (2^2) = \pi \cdot 4 = 4\pi \]
Thus, the area of the circle is:
\[ \boxed{4\pi} \]