Thank you for the clarification. If the radius is 6 cm (not the diameter), we'll need to recalculate everything accordingly.

Step 1: Correct the radius

• Given radius $r = 6 \, \text{cm}$.

Step 2: Area of the semicircle $\text{PAQ}$.

• The area of a semicircle is given by: $\text{Area of } \text{PAQ} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (6)^2 = \frac{36\pi}{2} = 18\pi \, \text{cm}^2$

Step 3: Area of the triangle $\text{POQ}$.

• The line segment $\text{PQ}$ is the base of the triangle, which is equal to the diameter of the semicircle.
• Since the radius is $6 \, \text{cm}$, the diameter $\text{PQ}$ is $12 \, \text{cm}$.
• The height of the triangle $\text{POQ}$ will be the radius $6 \, \text{cm}$.
• Thus, the area of the triangle $\text{POQ}$ is: $\text{Area of } \text{POQ} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times 6 = 36 \, \text{cm}^2$

Step 4: Area of the shaded region.

• The shaded region is the difference between the area of the semicircle $\text{PAQ}$ and the area of the triangle $\text{POQ}$: $\text{Area of shaded region} = \text{Area of } \text{PAQ} - \text{Area of } \text{POQ}$ Substituting the values: $\text{Area of shaded region} = 18\pi - 36 \, \text{cm}^2$

This is the exact area of the shaded region.

Would you like a numerical approximation or more detailed explanations?

Questions for Further Exploration:

1. How does the change from diameter to radius affect the semicircle's area?
2. If the radius was tripled, how would the shaded area change?
3. What is the ratio of the shaded region's area to the total area of the semicircle?
4. How would you calculate the area of the shaded region if the shape was a quarter circle instead?
5. What would be the effect of halving the radius on the area of the shaded region?
6. How would the problem change if the semicircle was inscribed in a square?
7. What would be the perimeter of the shaded region?
8. How can the concept of integration be used to verify the area of the shaded region?

Tip:

When dealing with areas involving semicircles and triangles, always verify the measurements of radius and diameter carefully, as they significantly impact the calculations.