Let's denote the side lengths of the first and second squares as $a$ and $b$, respectively.

Step 1: Find the side lengths of the squares

The perimeter of a square is given by $4 \times \text{side length}$. So, for the first square:

$4a = 40 \quad \Rightarrow \quad a = \frac{40}{4} = 10 \, \text{m}$

For the second square:

$4b = 96 \quad \Rightarrow \quad b = \frac{96}{4} = 24 \, \text{m}$

Step 2: Find the areas of the squares

The area of a square is given by the side length squared. Therefore, the area of the first square is:

$\text{Area of first square} = a^2 = 10^2 = 100 \, \text{m}^2$

The area of the second square is:

$\text{Area of second square} = b^2 = 24^2 = 576 \, \text{m}^2$

Step 3: Find the area of the new square

The problem asks for the perimeter of another square that has an area equal to the sum of the areas of the first two squares. Let's denote the side length of the new square as $c$.

The area of the new square is:

$c^2 = \text{Area of first square} + \text{Area of second square} = 100 + 576 = 676 \, \text{m}^2$

Step 4: Find the side length of the new square

Taking the square root of the area to find the side length $c$:

$c = \sqrt{676} = 26 \, \text{m}$

Step 5: Find the perimeter of the new square

The perimeter of the new square is:

$\text{Perimeter} = 4 \times c = 4 \times 26 = 104 \, \text{m}$

Thus, the perimeter of the square equal in area to the sum of the first two squares is 104 meters.

Would you like more details or have any questions?

Here are five related questions:

1. What is the area of the square with a perimeter of 104 meters?
2. How does the area of a square change if the perimeter is doubled?
3. What is the difference in area between two squares if one has a side length twice as long as the other?
4. How do you calculate the diagonal of a square given its side length?
5. What would be the perimeter of a square whose area is the difference between the areas of the two original squares?

Tip: When solving problems involving squares, remember that the relationship between the side length and the perimeter or area is quadratic. This can help in quickly checking your results.