https://file.aiquickdraw.com/mathbot/file/1726579041438txlprhq8.jpg

Let's break down the problem in the image to calculate the area of the blue-shaded region.

### Given:
- The figure consists of semicircles.
- \( AP = PQ = QB = 2 \, \text{cm} \).

This means:
- The total length \( AB \) is \( 6 \, \text{cm} \) (since \( AB = AP + PQ + QB = 2 + 2 + 2 = 6 \, \text{cm} \)).
- There are three semicircles in the figure.

### Steps to calculate the area of the blue-shaded region:
1. **Large semicircle with diameter \( AB \)**:
   - Diameter = \( AB = 6 \, \text{cm} \).
   - Radius \( R = \frac{AB}{2} = 3 \, \text{cm} \).
   - Area of this semicircle is:
   \[
   A_{\text{large semicircle}} = \frac{1}{2} \pi R^2 = \frac{1}{2} \pi (3)^2 = \frac{9\pi}{2} \, \text{cm}^2.
   \]

2. **Smaller semicircles**:
   - Each of these smaller semicircles has a diameter of \( 2 \, \text{cm} \), so the radius \( r = \frac{2}{2} = 1 \, \text{cm} \).
   
   - The two smaller semicircles are equal in size. The area of one such semicircle is:
   \[
   A_{\text{small semicircle}} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (1)^2 = \frac{\pi}{2} \, \text{cm}^2.
   \]
   
   - The total area of the two smaller semicircles is:
   \[
   A_{\text{two small semicircles}} = 2 \times \frac{\pi}{2} = \pi \, \text{cm}^2.
   \]

3. **Area of the blue-shaded region**:
   - The blue-shaded region is the area of the large semicircle minus the total area of the two smaller semicircles.
   \[
   A_{\text{blue region}} = A_{\text{large semicircle}} - A_{\text{two small semicircles}} = \frac{9\pi}{2} - \pi = \frac{7\pi}{2} \, \text{cm}^2.
   \]

Thus, the area of the blue-shaded region is \( \frac{7\pi}{2} \, \text{cm}^2 \), or approximately \( 10.995 \, \text{cm}^2 \).

Would you like to discuss any part of this solution in more detail?

### Related questions:
1. What would happen if the lengths \( AP \), \( PQ \), and \( QB \) were not equal?
2. How would the calculation change if the blue region were part of a quarter circle instead of semicircles?
3. What is the area of the non-shaded region in the diagram?
4. How can you generalize this problem for different semicircle configurations?
5. How would the calculation differ if the semicircles were complete circles instead of halves?

### Tip:
Always verify your understanding of geometric properties (such as areas and relationships between figures) before starting the calculation. This helps prevent common errors, especially with composite shapes.

What is the area of the region shaded in blue? The figures are made of semicircles and AP = PQ = QB = 2 cm.

Learn how to find the area of the blue-shaded region made of semicircles in this geometry problem. Given AP = PQ = QB = 2 cm, the total length AB is 6 cm, and the areas of the semicircles are calculated using standard geometry formulas. Suitable for students in Grades 9-10.

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