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To find the area of the shaded region, we need to calculate the area of the circular sector defined by the 102° angle, then subtract the area of the triangle formed within that sector.

Given:
- Radius of the circle, \( r = 9 \, \text{cm} \)
- Central angle of the sector, \( \theta = 102^\circ \)

### Step 1: Calculate the Area of the Sector
The formula for the area of a sector with angle \(\theta\) (in degrees) and radius \( r \) is:
\[
\text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2
\]
Substitute the values:
\[
\text{Area of sector} = \frac{102}{360} \times \pi \times 9^2
\]
\[
= \frac{102}{360} \times \pi \times 81
\]
\[
\approx 22.95 \pi \approx 72.12 \, \text{cm}^2
\]

### Step 2: Calculate the Area of the Triangle
To find the area of the triangle with two sides of 9 cm and an included angle of 102°, we use the formula:
\[
\text{Area of triangle} = \frac{1}{2} \times a \times b \times \sin(\theta)
\]
where \( a = 9 \), \( b = 9 \), and \( \theta = 102^\circ \).

\[
\text{Area of triangle} = \frac{1}{2} \times 9 \times 9 \times \sin(102^\circ)
\]
Using \(\sin(102^\circ) \approx 0.9781\),
\[
\text{Area of triangle} \approx \frac{1}{2} \times 81 \times 0.9781
\]
\[
\approx 39.7 \, \text{cm}^2
\]

### Step 3: Calculate the Area of the Shaded Region
The area of the shaded region is the area of the sector minus the area of the triangle:
\[
\text{Area of shaded region} = 72.12 - 39.7
\]
\[
\approx 32.42 \, \text{cm}^2
\]

### Final Answer:
The area of the shaded region is approximately \( \boxed{32.42 \, \text{cm}^2} \).

Would you like further details on any step? 

Here are some related questions to explore:
1. What is the formula for finding the area of a circular sector?
2. How do you find the area of a triangle given two sides and an included angle?
3. How would the area change if the angle was increased or decreased?
4. What is the effect on the area if the radius changes?
5. How does this approach change if given a central angle in radians?

**Tip:** In geometry problems, always double-check whether angles are in degrees or radians to apply the correct formulas.

The radius of the circle is 9 cm. Find the area of the shaded region in a circle with a central angle of 102°.

Find the area of the shaded region in a circle with a 9 cm radius and a 102° central angle. This geometry problem involves calculating the area of a circular sector and a triangle formed within, using trigonometric and geometric formulas. Ideal for students in Grades 9-10.

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