Let's analyze the diagram in the image step by step:

1. There is a square containing two circles. Each circle has a radius labeled: one circle has a radius of 1 meter, and the other has a radius of 3 meters.

2. These circles are inscribed within the square, meaning that the diameter of the circles plays a key role in defining the dimensions of the square.

Step 1: Calculate the diameters of the circles

• The diameter of a circle is twice the radius.
  • The larger circle's diameter = $2 \times 3 = 6$ meters.
  • The smaller circle's diameter = $2 \times 1 = 2$ meters.

Step 2: Analyze the configuration

Since the circles are inscribed in the square, the sum of the diameters of the two circles will equal one side of the square. Therefore, the side length of the square is: $\text{Side length of square} = 6 \, \text{m} + 2 \, \text{m} = 8 \, \text{m}.$

Step 3: Find the diagonal of the square

The diagonal of a square with side length $s$ can be found using the Pythagorean theorem: $\text{Diagonal} = s\sqrt{2}.$ Substituting $s = 8$: $\text{Diagonal} = 8\sqrt{2} \approx 8 \times 1.414 = 11.31 \, \text{meters}.$

Thus, the length of the diagonal of the square is approximately 11.31 meters.

Would you like a detailed breakdown of any of these steps or have further questions?

Here are five related questions you might find useful:

1. How would you find the area of the square given its side length?
2. What is the perimeter of the square based on the calculated side length?
3. How can you verify the placement of the circles inside the square using geometry?
4. How does the diagonal of the square relate to its side length geometrically?
5. What other shapes can be inscribed inside a square, and how would their properties compare?

Tip: Whenever dealing with a square, the diagonal always follows the relationship $\text{diagonal} = \text{side} \times \sqrt{2}$, which is useful in many geometry problems.