To find the area of the base of the hexagonal box, we use the formula for the area of a regular hexagon:

$A = \frac{3\sqrt{3}}{2} s^2$

where $s$ is the side length of the hexagon.

Step 1: Substitute $s = 9.5$ cm

$A = \frac{3\sqrt{3}}{2} (9.5)^2$

Step 2: Compute $s^2$

$9.5^2 = 90.25$

Step 3: Multiply by $\frac{3\sqrt{3}}{2}$

Approximating $\sqrt{3} \approx 1.732$:

$\frac{3 \times 1.732}{2} = \frac{5.196}{2} = 2.598$

$A \approx 2.598 \times 90.25$

$A \approx 234.43 \text{ cm}^2$

Final Answer:

$\mathbf{234.43 \text{ cm}^2}$

Would you like a breakdown of how the formula is derived?

Related Questions:

1. How do you derive the formula for the area of a regular hexagon?
2. What is the relationship between the hexagon’s side length and its height?
3. How does the area of a regular hexagon compare to an inscribed or circumscribed circle?
4. What is the perimeter of the hexagonal box base?
5. How would the area change if the side length were doubled?

Tip:

For any regular hexagon, you can also calculate the area using the formula:
$A = \frac{1}{2} \times \text{perimeter} \times \text{apothem}$
where the apothem is given as half the distance across opposite edges.