The ratio of the area of the inner square to the area of the outer square depends on the geometric relationship between the two squares. There are two common cases:

Case 1: Inner Square is Rotated Inside Outer Square

In this case, the inner square is rotated by 45 degrees inside the outer square, and its corners touch the midpoints of the sides of the outer square. If the side length of the outer square is $s$, the diagonal of the inner square is equal to the side length of the outer square. The side length of the inner square can be found using the diagonal relationship of a square: $d = s = \text{side of inner square} \times \sqrt{2}$ Thus, the side length of the inner square is: $\text{side of inner square} = \frac{s}{\sqrt{2}}$ The area of the inner square is: $A_{\text{inner}} = \left(\frac{s}{\sqrt{2}}\right)^2 = \frac{s^2}{2}$ The area of the outer square is: $A_{\text{outer}} = s^2$ So, the ratio of the area of the inner square to the area of the outer square is: $\frac{A_{\text{inner}}}{A_{\text{outer}}} = \frac{\frac{s^2}{2}}{s^2} = \frac{1}{2}$

Case 2: Inner Square is Inscribed within the Outer Square (Same Orientation)

If the inner square is inscribed in the outer square (same orientation, with parallel sides), the side lengths of the two squares are directly proportional. If the side length of the inner square is $a$ and the side length of the outer square is $s$, the ratio of the areas is simply: $\frac{A_{\text{inner}}}{A_{\text{outer}}} = \frac{a^2}{s^2}$

Would you like details or have any further questions?

Here are 5 related questions to expand the concept:

1. How is the area affected when scaling the side lengths of a square?
2. What is the perimeter ratio between the inner and outer square in both cases?
3. How would the ratio change if the inner shape were a different polygon?
4. How does the ratio of areas change if the outer square is scaled by a factor of 2?
5. What is the relationship between the diagonals of both squares?

Tip: When working with squares, diagonal relationships often involve the factor $\sqrt{2}$, especially when dealing with rotations.