To find the area of a square with a given diagonal, we can use the relationship between the side length of the square and its diagonal.

1. Let $s$ be the side length of the square.
2. For a square, the diagonal $d$ is related to the side length by the formula: $d = s\sqrt{2}$
3. In this case, the diagonal $d$ is 40 meters, so: $40 = s\sqrt{2}$
4. Solving for $s$: $s = \frac{40}{\sqrt{2}} = 40 \times \frac{\sqrt{2}}{2} = 20\sqrt{2}$
5. Now, the area $A$ of the square is given by $A = s^2$: $A = (20\sqrt{2})^2 = 20^2 \times 2 = 400 \times 2 = 800 \, \text{square meters}$

Answer:

The area of the square is $800$ square meters.

Would you like further details on any part of the solution or have any other questions?

Here are some related questions to explore:

1. What would the side length of a square be if its area is 800 square meters?
2. How would you find the perimeter of a square given its diagonal?
3. If the diagonal of a square was doubled, how would the area change?
4. What is the relationship between the diagonal and side length of other regular polygons?
5. How does changing the diagonal of a square affect its overall geometry?

Tip: Knowing the relationship between the diagonal and side of a square is helpful, as many problems involving squares provide the diagonal rather than the side length directly.