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To solve this problem, we need to calculate the area of the shaded region in the circular sector, given:

- \( \theta = 30^\circ \)
- \( r = 6 \, \text{cm} \) (radius of the circle)

### Step 1: Calculate the area of the sector
The area of a circular sector is given by the formula:
\[
A_{\text{sector}} = \frac{\theta}{360^\circ} \times \pi r^2
\]
Substituting the values:
\[
A_{\text{sector}} = \frac{30^\circ}{360^\circ} \times \pi \times 6^2 = \frac{1}{12} \times \pi \times 36 = 3\pi \, \text{cm}^2
\]

### Step 2: Calculate the area of the triangle
The sector forms an isosceles triangle with two sides equal to the radius \( r = 6 \, \text{cm} \) and the angle \( \theta = 30^\circ \). To calculate the area of this triangle, we can use the formula for the area of a triangle with two sides \( a \) and \( b \) and the included angle \( \theta \):
\[
A_{\text{triangle}} = \frac{1}{2} a b \sin(\theta)
\]
Since both sides are equal to the radius, \( a = b = 6 \, \text{cm} \), and \( \theta = 30^\circ \):
\[
A_{\text{triangle}} = \frac{1}{2} \times 6 \times 6 \times \sin(30^\circ) = \frac{1}{2} \times 36 \times \frac{1}{2} = 9 \, \text{cm}^2
\]

### Step 3: Calculate the area of the shaded region
The shaded region is the area of the sector minus the area of the triangle:
\[
A_{\text{shaded}} = A_{\text{sector}} - A_{\text{triangle}} = 3\pi - 9 \, \text{cm}^2
\]

### Step 4: Choose the correct answer
The expression for the area of the shaded region matches option **A**: \( 3(\pi - 9) \).

Thus, the correct answer is **A**.

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Do you have any questions or need further clarification? Here are 5 related questions you can explore:

1. How do you find the area of a sector for a general angle and radius?
2. What is the formula for the area of a triangle using trigonometry?
3. How can you convert degrees to radians, and when would you need that?
4. How do you calculate the circumference of a circle?
5. What is the relationship between a sector's area and the whole circle's area?

**Tip:** Always check if the angle given is in degrees or radians when working with sector areas or trigonometry!

¿Cuál es el valor del área de la región sombreada? El ángulo θ del sector circular mide 30°, mientras que el radio r mide 6 cm.

Learn how to calculate the shaded area in a circular sector where the angle is 30° and the radius is 6 cm. This step-by-step solution covers the area of a sector and an isosceles triangle using trigonometric formulas. Suitable for Grades 9-11.

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