https://file.aiquickdraw.com/mathbot/file/172637499080467n7xkyk.jpg

To find the area of the shaded segment in the given circle, we need to use the formula for the area of a segment. The area of the shaded segment is the difference between the area of the sector \( A_{\text{sector}} \) and the area of the triangle \( A_{\text{triangle}} \).

### Step 1: Area of the sector
The area of a sector is given by:
\[
A_{\text{sector}} = \frac{1}{2} r^2 \theta
\]
where \( r \) is the radius of the circle, and \( \theta \) is the angle in radians.

From the image:
- \( r = 1.6 \, \text{m} \)
- \( \theta = \frac{4\pi}{9} \, \text{radians} \)

Substituting these values:
\[
A_{\text{sector}} = \frac{1}{2} \times (1.6)^2 \times \frac{4\pi}{9}
\]

### Step 2: Area of the triangle
The area of the triangle can be calculated using the formula:
\[
A_{\text{triangle}} = \frac{1}{2} r^2 \sin(\theta)
\]
Substitute the known values:
\[
A_{\text{triangle}} = \frac{1}{2} \times (1.6)^2 \times \sin\left(\frac{4\pi}{9}\right)
\]

### Step 3: Area of the shaded segment
The area of the shaded segment is:
\[
A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}}
\]

I will now calculate these areas and find the final answer.The area of the shaded segment is approximately \( 0.527 \, \text{m}^2 \), correct to 3 significant figures.

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. How would the formula change if the angle was given in degrees instead of radians?
2. What is the significance of using radians in calculations for circular areas?
3. How can you find the area of a sector when only the arc length is given?
4. How does increasing the radius affect the area of the shaded segment?
5. How would you calculate the perimeter of the shaded segment?

**Tip:** For accurate calculations in geometry, always ensure that angles are in the correct units (radians or degrees) based on the formulas you're using.

The shaded segment is formed by an arc BC of a circle with radius 1.6m, subtending an angle of 4π/9 radians at its center A. Find the area of the shaded segment, correct to 3 significant figures.

Learn how to calculate the area of a shaded segment formed by a 1.6m radius circle with a subtended angle of 4π/9 radians. Step-by-step solution using the sector area formula and trigonometric triangle area. Suitable for high school students in Grades 9-12.

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