The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.
Given that a circle with a radius of \( 2 \) inches has an area of \( 12.568 \) square inches, we can confirm that the area does indeed vary directly as the square of the radius by checking if the formula holds true.
Let's verify it:
\[ A = \pi(2^2) = \pi(4) \approx 12.5664 , \text{in}^2 \]
Since \( 12.568 \) is very close to \( \pi(4) \), we can conclude that \( \pi \approx 3.1416 \) aligns with the calculation.
Now, we need to find the area of a circle with a radius of \( 5.7 \) inches:
\[ A = \pi (5.7^2) = \pi (32.49) \]
Calculating \( \pi (32.49) \):
\[ A \approx 3.1416 \times 32.49 \approx 102.192 , \text{in}^2 \]
Thus, the area of a circle with a radius of \( 5.7 \) inches is approximately:
\[ \boxed{102.192} , \text{in}^2 \]