To solve this problem, we need to find the area of the shaded region, which is the difference between the area of the square and the area of the circle placed inside it.

Step 1: Area of the square

The side length of the square is given as 8 mm.

The formula for the area of a square is: $A_{\text{square}} = \text{side}^2$

So, the area of the square is: $A_{\text{square}} = 8^2 = 64 \text{ mm}^2$

Step 2: Area of the circle

The circle is inscribed in the square, meaning that its diameter is equal to the side length of the square. Therefore, the diameter of the circle is 8 mm.

The radius $r$ of the circle is half the diameter: $r = \frac{8}{2} = 4 \text{ mm}$

The formula for the area of a circle is: $A_{\text{circle}} = \pi r^2$

Substitute $r = 4 \text{ mm}$: $A_{\text{circle}} = \pi (4)^2 = 16\pi \text{ mm}^2$

Using an approximation for $\pi \approx 3.1416$, we get: $A_{\text{circle}} \approx 16 \times 3.1416 = 50.2656 \text{ mm}^2$

Step 3: Area of the shaded region

The shaded region is the area of the square minus the area of the circle: $A_{\text{shaded}} = A_{\text{square}} - A_{\text{circle}}$

Substitute the values we found: $A_{\text{shaded}} = 64 - 50.2656 = 13.7344 \text{ mm}^2$

Final Answer:

The area of the shaded region is approximately $13.7344 \text{ mm}^2$.

Would you like more details or clarification on any step?

Here are some related questions you might find interesting:

1. What is the area of a circle with a diameter of 10 mm?
2. If the square had a side length of 12 mm, what would be the new shaded area?
3. How would the problem change if the circle was circumscribed outside the square?
4. What is the perimeter of the square in this problem?
5. Can you find the circumference of the inscribed circle?

Tip: When working with circles, it's useful to remember that the radius is always half of the diameter!