To find the area of the regular hexagon, we use the formula for the area of a regular polygon:

$\text{Area} = \frac{3\sqrt{3}}{2} \times s^2$

Where $s$ is the side length of the hexagon. Given that the distance from the center to any vertex is 10 km, we know this is the radius, which is also the side length of the hexagon.

So, we plug in $s = 10 \, \text{km}$:

$\text{Area} = \frac{3\sqrt{3}}{2} \times (10)^2$

Now, let's compute the area:

$\text{Area} = \frac{3\sqrt{3}}{2} \times 100 = \frac{3 \times 1.732}{2} \times 100 \approx 259.8 \, \text{square kilometers}$

So, the area of the hexagon is approximately 259.8 square kilometers.

Would you like further details on this calculation, or do you have any other questions?

Here are some related questions to help you explore more:

1. How would the area change if the radius were doubled?
2. What is the relationship between the number of sides and the area of a regular polygon?
3. How can you find the area of a regular polygon with a known apothem instead of a side length?
4. How do you calculate the perimeter of a regular hexagon?
5. What is the formula for the area of a triangle within a regular hexagon?

Tip: When working with regular polygons, knowing the relationship between the radius and the side length is crucial, as it simplifies many area and perimeter calculations.