Step 1: Pick a convenient actual radius rather than working symbolically.
Since we will use $\pi = \frac{22}{7}$, let the circle's radius be $r = 7$ cm, a multiple of 7 keeps the numbers clean.
Step 2: Find the circle's circumference, and match the square's perimeter to it. \[ C = 2 \times \frac{22}{7} \times 7 = 44 \text{ cm} \] Since the square has the same perimeter, its side is \[ a = \frac{44}{4} = 11 \text{ cm} \]
Step 3: Work out both areas with these actual numbers. \[ \text{Area of circle} = \frac{22}{7} \times 7^2 = \frac{22}{7} \times 49 = 154 \text{ cm}^2 \] \[ \text{Area of square} = 11^2 = 121 \text{ cm}^2 \]
Step 4: Write the ratio in simplest form. \[ 154 : 121 = 14 : 11 \] \[ \boxed{14 : 11} \]