https://file.aiquickdraw.com/mathbot/file/1723304049506s6nka9tb.jpg

To find the area of the shaded region, let's break down the problem step by step.

### Description of the Diagram:
- The diagram shows a semicircle \( \text{PAQ} \) with diameter \( \text{OA} = 6 \, \text{cm} \). 
- \( \text{B} \) is a point on the semicircle \( \text{PAQ} \) such that \( \text{BO} \) and \( \text{BA} \) form two lines intersecting at point \( O \).
- The shaded region is the area between the line \( \text{PQ} \) and the arc \( \text{PQ} \).

### Step 1: Calculate the radius of the semicircle.
The diameter \( \text{OA} = 6 \, \text{cm} \), so the radius \( r \) is:
\[
r = \frac{\text{OA}}{2} = \frac{6}{2} = 3 \, \text{cm}
\]

### Step 2: Area of the semicircle \( \text{PAQ} \).
The area of a semicircle is given by:
\[
\text{Area of } \text{PAQ} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (3)^2 = \frac{9\pi}{2} \, \text{cm}^2
\]

### Step 3: Area of the triangle \( \text{POQ} \).
The line segment \( \text{PQ} \) is the diameter of a smaller semicircle. Since \( \text{OA} = 6 \, \text{cm} \), and \( \text{O} \) is the midpoint, the length \( \text{PQ} \) is also \( 6 \, \text{cm} \).
The height of the triangle \( \text{POQ} \) will be equal to the radius of the semicircle, which is \( 3 \, \text{cm} \). Thus, the area of triangle \( \text{POQ} \) is:
\[
\text{Area of } \text{POQ} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 3 = 9 \, \text{cm}^2
\]

### 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:
\[
\text{Area of shaded region} = \frac{9\pi}{2} - 9 \, \text{cm}^2
\]
This expression gives the exact area of the shaded region.

If you need further details or a specific numerical approximation, please let me know!

### Questions for Further Exploration:
1. What is the area of the full circle if the diameter is doubled?
2. How would the area of the shaded region change if the radius were halved?
3. What is the relationship between the area of a circle and its radius?
4. Can the area of the shaded region be expressed as a fraction of the area of the semicircle?
5. How does the area of the triangle \( \text{POQ} \) compare to the area of the shaded region?
6. If the triangle \( \text{POQ} \) were equilateral, how would the problem change?
7. How would you find the area of the shaded region if the triangle was not right-angled?
8. How does changing the diameter affect the length of the arc \( \text{PQ} \)?

### Tip:
Always ensure to double-check each calculation step when dealing with composite geometric figures, as small errors can significantly impact the final result.

Learn how to find the area of the shaded region in a semicircle problem where OA = 6 cm. Step-by-step solution includes calculating the area of a semicircle and the area of a triangle. Suitable for students in Grades 7-9.

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