To find the area of the shaded region in the given image, we need to subtract the area of the inscribed rectangle from the area of the circle.

Here are the steps:

1. Area of the circle:

   • The diagonal of the rectangle (AC) is the diameter of the circle.
   • To find the diagonal, use the Pythagorean theorem: $\text{Diagonal (AC)} = \sqrt{(8 \, \text{cm})^2 + (6 \, \text{cm})^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \, \text{cm}$
   • So, the diameter of the circle is 10 cm, making the radius $\frac{10}{2} = 5 \, \text{cm}$.
   • The area of the circle is given by: $\text{Area of circle} = \pi r^2 = \pi (5)^2 = 25\pi \, \text{cm}^2$

2. Area of the rectangle:

   • The dimensions of the rectangle are given as 8 cm by 6 cm.
   • The area of the rectangle is: $\text{Area of rectangle} = 8 \times 6 = 48 \, \text{cm}^2$

3. Shaded area:

   • The shaded area is the area of the circle minus the area of the rectangle: $\text{Shaded area} = 25\pi - 48 \, \text{cm}^2$
   • Using $\pi \approx 3.1416$: $\text{Shaded area} \approx 25(3.1416) - 48 = 78.54 - 48 = 30.54 \, \text{cm}^2$

Thus, the area of the shaded region is approximately $30.54 \, \text{cm}^2$.

Would you like more details on this, or do you have any questions?

Here are some related questions for deeper understanding:

1. How do we derive the formula for the area of a circle?
2. What is the significance of the Pythagorean theorem in geometry?
3. Can you calculate the perimeter of the rectangle in the figure?
4. How would the shaded area change if the dimensions of the rectangle were different?
5. What happens to the area of the circle if its radius is doubled?

Tip: Always double-check your use of units in geometry to avoid confusion, especially when combining different measurements.