To solve this problem, we need to determine the area of the square yard inside the fence.

Step 1: Determine the perimeter of the square yard

The problem states that the fence is built with 48 m² of lumber and the fence is 1.5 meters tall. Since the total lumber area is the product of the fence's height and the perimeter of the square yard:

$\text{Perimeter of the yard} = \frac{\text{Total lumber area}}{\text{Height of the fence}}$

Substituting the values:

$\text{Perimeter} = \frac{48}{1.5} = 32 \, \text{meters}.$

Step 2: Calculate the side length of the square yard

For a square, the perimeter is given by:

$\text{Perimeter} = 4 \times \text{Side length}.$

Using the calculated perimeter:

$32 = 4 \times \text{Side length}.$

$\text{Side length} = \frac{32}{4} = 8 \, \text{meters}.$

Step 3: Calculate the area of the square yard

The area of a square is given by:

$\text{Area} = \text{Side length}^2.$

Substituting the side length:

$\text{Area} = 8^2 = 64 \, \text{m}^2.$

Final Answer:

The area of the yard inside the fence is:

$\boxed{64 \, \text{m}^2}.$

Would you like a detailed breakdown of any step or further clarifications?

Here are 5 follow-up questions for you:

1. How does the height of the fence relate to the total lumber used?
2. What if the yard was rectangular instead of square—how would the calculations change?
3. Can you determine the cost of the lumber if the cost per square meter is provided?
4. What other shapes (e.g., circular) could use the same amount of lumber, and how would their areas compare?
5. How would the area change if the perimeter increased but the shape remained square?

Tip: Always ensure measurements are consistent in units before solving geometry problems!