To calculate the area of a sector of a circle, we use the formula:

$\text{Area of sector} = \frac{1}{2} r^2 \theta$

Where:

• $r$ is the radius of the circle.
• $\theta$ is the angle subtended by the sector in radians.

In this case:

• $r = 4 \, \text{km}$
• $\theta = \frac{5\pi}{8} \, \text{radians}$

Now, substituting these values into the formula:

$\text{Area of sector} = \frac{1}{2} \times (4^2) \times \frac{5\pi}{8}$

Simplifying:

$\text{Area of sector} = \frac{1}{2} \times 16 \times \frac{5\pi}{8} = 8 \times \frac{5\pi}{8} = 5\pi \, \text{km}^2$

Thus, the area of the sector is:

$5\pi \, \text{km}^2 \approx 15.71 \, \text{km}^2$

Would you like more details or further clarification?

Here are 5 related questions to explore further:

1. What is the perimeter of the sector, including the arc length?
2. How do we calculate the arc length of a sector?
3. What would the area of the sector be if the angle was in degrees?
4. How does changing the radius affect the area of the sector?
5. What is the total area of the circle from which the sector was derived?

Tip: Always ensure the angle is in radians when using the formula for the area of a sector, as this is the standard form.