Correct Answer: 6 : 11

Solution : Area of the square = 1296 cm$^2$
Radius of the circle = $\frac{7}{6}$ of the length of a side of the square
Area of the square = Side$^2$ = 1296
Side = $\sqrt{1296}$ = 36 cm
Now, Radius of circle = $\frac{7}{6}\times 36$ = 42 cm
Perimeter of the square = 4 × Side
Perimeter of the square = 4 × 36 = 144 cm
Circumference of the circle = $2\pi r$
Circumference of circle = $2\times \frac{22}{7} \times 42 = 44 \times 6$  = 264 cm
The ratio of the perimeter of the square and the circumference of the circle = $\frac{144}{264}= \frac{6}{11}$ = 6 : 11
$\therefore$ The ratio of the perimeter of the square and the circumference of the circle is 6 : 11.
Hence, the correct answer is 6 : 11.