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The problem in the image asks to find the area of the shaded region in a circle. Here's how we approach it step by step:

### 1. Identify the key elements:
- **Radius (r) = 3** units.
- **Central angle (θ) = 120°**.
- The shaded area is the part of the circle corresponding to this angle minus the triangular region formed by the two radii and the chord.

### 2. Calculate the area of the sector:
The formula for the area of a sector of a circle is:
\[
\text{Area of sector} = \frac{\theta}{360} \times \pi r^2
\]
Substitute the given values:
\[
\text{Area of sector} = \frac{120}{360} \times \pi \times 3^2 = \frac{1}{3} \times \pi \times 9 = 3\pi \, \text{square units}
\]
\[
\text{Area of sector} \approx 3 \times 3.1416 = 9.4248 \, \text{square units}
\]

### 3. Calculate the area of the triangle:
The triangle formed is an **isosceles triangle**, and we can find its area using the formula for the area of a triangle:
\[
\text{Area of triangle} = \frac{1}{2} r^2 \sin(\theta)
\]
Substituting the given values:
\[
\text{Area of triangle} = \frac{1}{2} \times 3^2 \times \sin(120^\circ)
\]
\[
= \frac{1}{2} \times 9 \times \sin(120^\circ)
\]
We know that \(\sin(120^\circ) = \sin(60^\circ) = \frac{\sqrt{3}}{2}\), so:
\[
\text{Area of triangle} = \frac{1}{2} \times 9 \times \frac{\sqrt{3}}{2} = \frac{9\sqrt{3}}{4} \approx 3.8971 \, \text{square units}
\]

### 4. Find the area of the shaded region:
The shaded region is the difference between the area of the sector and the area of the triangle:
\[
\text{Area of shaded region} = \text{Area of sector} - \text{Area of triangle}
\]
\[
\text{Area of shaded region} \approx 9.4248 - 3.8971 = 5.5277 \, \text{square units}
\]

### Final answer:
The area of the shaded region is approximately **5.53 square units**.

Would you like any more details or explanations?

Here are five related questions:
1. How do we calculate the area of a sector for different central angles?
2. How does the sine rule help in calculating areas of triangles with non-right angles?
3. How does the central angle affect the area of the shaded region in a circle?
4. What is the relationship between radians and degrees in circular measurements?
5. How can you find the area of a circular segment without using trigonometry?

**Tip:** Always make sure your calculator is in the correct mode (degrees or radians) when calculating trigonometric functions based on angles!

Find the area of the shaded region in the circle with a central angle of 120° and a radius of 3 units.

Learn how to find the area of the shaded region in a circle with a central angle of 120° and a radius of 3 units. This detailed step-by-step solution involves calculating the area of the sector and subtracting the area of the isosceles triangle. Perfect for students in Grades 10-12.

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