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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 \( 4 \times s \).

Would

The diagram shows four identical circles drawn inside a square. Each circle touches two other circles and two sides of the square. The region inside the square that is outside the circles, shown shaded in the diagram, has a total area of 40 cm². Work out the perimeter of the square. Give your answer correct to 3 significant figures.

Solve a geometry problem involving four identical circles inside a square with a shaded area of 40 cm² outside the circles. Learn how to determine the perimeter of the square using algebraic manipulation and area calculations. Suitable for high school students in Grades 10-12.

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