To solve this problem, let's break it down step by step.

1. Area of the Outer Square

The area of any square is given by the formula:

$\text{Area of square} = \text{side}^2$

Let’s say the side length of the outer square is $s_{\text{outer}}$. So the area of the outer square will be:

$\text{Area of outer square} = s_{\text{outer}}^2$

2. Area of the 4 Triangles

Next, we need to find the area of the four triangles. To do this, we need the dimensions of the triangles, typically the base and the height.

If we assume that each triangle has a base $b$ and height $h$, the area of one triangle is:

$\text{Area of one triangle} = \frac{1}{2} \times b \times h$

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

$\text{Area of 4 triangles} = 4 \times \left( \frac{1}{2} \times b \times h \right) = 2 \times b \times h$

3. Area of the Inner Square

The area of the inner square can now be found by subtracting the area of the four triangles from the area of the outer square:

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

Substituting the earlier expressions:

$\text{Area of inner square} = s_{\text{outer}}^2 - 2 \times b \times h$

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To find the exact area of the inner square, we would need specific values for the side length of the outer square and the dimensions of the triangles.

Let me know if you have those details or if you'd like further clarification on any of the steps!

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Here are a few follow-up questions to explore more:

1. What are the dimensions of the outer square and the triangles?
2. How do the angles of the triangles relate to the dimensions of the inner square?
3. What would happen if the triangles were right-angled? How would this change the area formula?
4. Could this setup be generalized for different polygons, and how?
5. How would the area change if the number of triangles was increased or decreased?

Tip: Always double-check if there’s any specific information about the triangles or square that might simplify the calculations (like symmetry or special angles).