The problem you're referring to is asking you to find the area of the colored part of the circle. The square inside the circle has a side length of 8 cm.

Here's how to approach the problem:

1. Step 1: Area of the square

   • The area of a square is given by $A_{\text{square}} = \text{side}^2$.
   • Since the side of the square is 8 cm, the area of the square is: $A_{\text{square}} = 8^2 = 64 \, \text{cm}^2$

2. Step 2: Determine the radius of the circle

   • The square is inscribed in the circle, meaning the diameter of the circle is equal to the diagonal of the square.
   • The diagonal $d$ of a square can be calculated using Pythagoras' theorem: $d = \sqrt{8^2 + 8^2} = \sqrt{64 + 64} = \sqrt{128} = 8\sqrt{2} \, \text{cm}$
   • The radius $r$ of the circle is half of the diagonal: $r = \frac{8\sqrt{2}}{2} = 4\sqrt{2} \, \text{cm}$

3. Step 3: Area of the circle

   • The area of a circle is given by $A_{\text{circle}} = \pi r^2$.
   • Substituting $r = 4\sqrt{2}$: $A_{\text{circle}} = \pi (4\sqrt{2})^2 = \pi \times 32 \approx 100.53 \, \text{cm}^2$

4. Step 4: Area of the colored region

   • The colored region is the area of the circle minus the area of the square: $A_{\text{colored}} = A_{\text{circle}} - A_{\text{square}} = 100.53 - 64 = 36.53 \, \text{cm}^2$

Final Answer:

The area of the colored part of the circle is approximately 36.53 cm².

Would you like further clarification or a breakdown of any specific step?

Here are some related questions for practice:

1. If the side of the square was doubled, what would the area of the colored part be?
2. How does the area of the circle change if the side length of the square is reduced by half?
3. What is the relationship between the diagonal of a square and the radius of the circle it inscribes?
4. Can you find the perimeter of the square and the circumference of the circle?
5. How would the answer change if the square was circumscribed outside the circle instead?

Tip: Always check if shapes are inscribed or circumscribed to understand their dimensional relationships.