Step 1: Define Variables and Equations.
Let the radius of the circle be \( r \) and the side length of the square be \( s \).
Perimeter of the circle: \( 2\pi r \).
Perimeter of the square: \( 4s \).
Given that the perimeters are equal:
\[
2\pi r = 4s.
\]
Solve for \( s \) in terms of \( r \):
\[
s = \frac{\pi r}{2}.
\]
Step 2: Calculate the Areas.
Area of the circle: \( \pi r^2 \).
Area of the square:
\[
s^2 = \left(\frac{\pi r}{2}\right)^2 = \frac{\pi^2 r^2}{4}.
\]
Step 3: Find the Ratio of Areas.
The ratio of the area of the circle to the area of the square is:
\[
\text{Ratio} = \frac{\text{Area of circle}}{\text{Area of square}} = \frac{\pi r^2}{\frac{\pi^2 r^2}{4}} = \frac{\pi r^2 \cdot 4}{\pi^2 r^2} = \frac{4}{\pi}.
\]
To express this ratio in terms of integers, approximate \( \pi \approx \frac{22}{7} \):
\[
\frac{4}{\pi} \approx \frac{4}{\frac{22}{7}} = \frac{4 \times 7}{22} = \frac{28}{22} = \frac{14}{11}.
\]
Thus, the ratio of the areas is \( 14:11 \).
Step 4: Analyze the Options.
Option (1): \( 11:14 \) — Incorrect, as this is the reciprocal of the correct ratio.
Option (2): \( 5:6 \) — Incorrect, as this does not match the calculated value.
Option (3): \( 14:11 \) — Correct, as it matches the calculated value.
Option (4): \( 7:8 \) — Incorrect, as this does not match the calculated value.
Step 5: Final Answer.
\[
(3) \quad \mathbf{14:11}
\]