To find the area of the inner square, we can follow these steps:

Step 1: Find the side length of the outer square

The outer square has sides that are divided into two parts by the inner square. Each side of the outer square is divided into two segments, one of length 3 units and the other of length 2 units. Therefore, the total length of each side of the outer square is:

$\text{Side length of outer square} = 3 + 2 = 5 \text{ units}$

Step 2: Find the area of the outer square

The area $A_{\text{outer}}$ of a square is given by the square of the side length. So for the outer square:

$A_{\text{outer}} = \text{Side length}^2 = 5^2 = 25 \text{ square units}$

Step 3: Find the area of the 4 triangles

Each of the 4 triangles formed by the inner square has a base of 3 units and a height of 2 units. The area of one triangle $A_{\text{triangle}}$ is:

$A_{\text{triangle}} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 3 \times 2 = 3 \text{ square units}$

Since there are 4 triangles, the total area of the 4 triangles is:

$\text{Total area of triangles} = 4 \times 3 = 12 \text{ square units}$

Step 4: Find the area of the inner square

Now, we can use the formula given in the problem:

$\text{Area of inner square} = \text{Area of outer square} - \text{Total area of the 4 triangles}$

Substituting the values we found:

$\text{Area of inner square} = 25 - 12 = 13 \text{ square units}$

Final Answer:

The area of the inner square is 13 square units.

Would you like more details on any of the steps, or do you have any questions about this solution?

Here are 5 related questions to explore further:

1. How do we find the side length of the inner square from its area?
2. What is the relationship between the areas of the triangles and the squares?
3. How can we calculate the diagonal length of the inner square?
4. Can the same method be applied to find areas when the inner square is not tilted?
5. What would happen if the inner square was rotated differently inside the outer square?

Tip: When working with geometric shapes, drawing the problem out can often help visualize the relationships between the different components.