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.