To solve this problem, we need to find the area of the colored part of a circle that is inscribed in a square with a side length of 8 cm.

Step-by-Step Solution

1. Find the Area of the Square:

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

2. Find the Radius of the Inscribed Circle:

   • A circle inscribed in a square will have its diameter equal to the side length of the square.
   • Therefore, the diameter of the circle is also 8 cm.
   • The radius $r$ of the circle is half the diameter: $r = \frac{\text{diameter}}{2} = \frac{8}{2} = 4 \, \text{cm}$

3. Find the Area of the Circle:

   • The area of a circle ($A_{\text{circle}}$) is given by: $A_{\text{circle}} = \pi r^2 = \pi \times 4^2 = 16\pi \, \text{cm}^2$

4. Find the Area of the Colored Part:

   • If we assume the "colored part" refers to the area inside the circle but outside the square, we need to calculate the difference between the circle's area and the square's area: $A_{\text{colored part}} = A_{\text{circle}} - A_{\text{square}} = 16\pi - 64 \, \text{cm}^2$

Final Answer

The area of the colored part of the circle is: $A_{\text{colored part}} = 16\pi - 64 \, \text{cm}^2 \approx 50.27 \, \text{cm}^2 \quad (\text{using} \, \pi \approx 3.14)$

Would you like more details on any part of this solution, or do you have any other questions?

Related Questions to Explore

1. How would the area of the colored part change if the side length of the square were doubled?
2. What is the perimeter of the circle and the square in the problem?
3. How would you calculate the area of the shaded region if only part of the circle were colored?
4. How does the area of the colored part change if the circle is not inscribed but circumscribed around the square?
5. Can you find the ratio of the area of the circle to the area of the square?

Tip

Remember that when dealing with geometric shapes inscribed within each other, their properties (like side lengths and radii) often relate directly, providing useful shortcuts in solving area and perimeter problems.