The problem asks to find the length of the diagonal of a square, which contains two circles with a radius of 1 meter each, and these circles are positioned inside the square in a way that their edges touch the sides of the square and each other.

Let's break down the problem:

1. The diameter of each circle is $2 \times 1 = 2 \, \text{meters}$.

2. The two circles are arranged in such a way that:

   • They are touching each other along one side.
   • They touch two adjacent sides of the square.

This means that the side of the square will be equal to the sum of the diameters of the two circles, since the circles fit perfectly along the side.

$\text{Side of the square} = 2 \, \text{m} + 2 \, \text{m} = 4 \, \text{m}$

Finding the diagonal:

For a square, the length of the diagonal can be found using the Pythagorean theorem: $d = s\sqrt{2}$ where $s$ is the side length of the square.

In this case: $d = 4\sqrt{2} \, \text{m}$

Now, let's calculate the exact value of the diagonal length.

$d = 4 \times \sqrt{2} \approx 4 \times 1.414 = 5.656 \, \text{meters}$

Conclusion:

The length of the diagonal of the square is approximately 5.656 meters.

Would you like further details or explanations?

Here are five related questions for deeper understanding:

1. How is the Pythagorean theorem used to find the diagonal of a square?
2. What if the circles had different radii—how would the side of the square change?
3. How would the diagonal change if the radius of each circle doubled?
4. What formula would you use if the shape were a rectangle instead of a square?
5. How can you calculate the area of the square given its diagonal?

Tip: For any square, the relationship between the side and diagonal is always $d = s\sqrt{2}$, a helpful formula to remember!