To solve this problem, we need to find the perimeter of the square given that the shaded area (the area outside the circles but within the square) is $40 \, \text{cm}^2$.

Step-by-Step Solution

1. Define Variables and Geometry:

   • Let $s$ be the side length of the square.
   • Let $r$ be the radius of each circle.

2. Area of the Square:

   • The area of the square, $A_{\text{square}}$, is $s^2$.

3. Area of the Circles:

   • Since there are four identical circles, the total area of all the circles is $4 \times \pi r^2 = 4\pi r^2$.

4. Shaded Area:

   • According to the problem, the shaded area (area of the square minus the area covered by the circles) is $40 \, \text{cm}^2$.
   • So, we have: $s^2 - 4\pi r^2 = 40$

5. Relationship Between $s$ and $r$:

   • Each circle touches two sides of the square, meaning the diameter of each circle is equal to half of the side length of the square.
   • Therefore, $2r = \frac{s}{2}$ or $s = 4r$.

6. Substitute $s = 4r$ into the Shaded Area Equation:

   • Replace $s$ in the equation: $(4r)^2 - 4\pi r^2 = 40$ $16r^2 - 4\pi r^2 = 40$

7. Solve for $r$:

   • Factor out $r^2$ from the equation: $r^2(16 - 4\pi) = 40$
   • Divide both sides by $16 - 4\pi$: $r^2 = \frac{40}{16 - 4\pi}$
   • Calculate $r^2$ and then $r$ using the approximation $\pi \approx 3.14159$.

8. Calculate the Side Length $s$:

   • Substitute $r$ back to find $s = 4r$.

9. Calculate the Perimeter of the Square:

   • The perimeter $P$ of the square is $4 \times s$.

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