To calculate the volume of the hexagonal prism, we can use the formula for the volume of a prism:
\[ \text{Volume} = \text{Base Area} \times \text{Height} \]
Step 1: Determine the base area of the hexagonal prism
The base of the hexagonal prism is a regular hexagon. The area \(A\) of a regular hexagon can be calculated using the formula:
\[ A = \frac{3\sqrt{3}}{2} s^2 \]
where \(s\) is the length of a side of the hexagon.
From the information provided, we know that the two longer parallel sides measure 30 cm each. This means that these are the lengths of two opposite sides of the hexagon. For a regular hexagon, each side has the same length. Therefore, the length of each side (s) can be determined from the relationship of the hexagon's geometry.
A regular hexagon can be divided into 6 equilateral triangles. The length of the side of an equilateral triangle can be calculated as follows:
To find the height of the trapezoid formed by the two parallel sides of the hexagon, we can draw a perpendicular line from the midpoint of one parallel side to the midpoint of the other, which is given by \(15\) cm.
Step 2: Calculate the base's geometry
By geometry, the equation from the hexagon is:
\[ s = \frac{\text{Length of one side}}{2} = 15 , \text{cm} \]
Step 3: Calculate the area of the hexagonal base
Using \(s = 15 cm\):
\[ A = \frac{3\sqrt{3}}{2} (15)^2 = \frac{3\sqrt{3}}{2}(225) = 337.5\sqrt{3} \approx 583.1 , \text{cm}^2 \]
Step 4: Calculate the volume
With the height of the prism being \(8 , \text{cm}\):
\[ \text{Volume} = \text{Base Area} \times \text{Height} = 337.5\sqrt{3} \times 8 \]
Substituting the approximate value gives:
\[ \text{Volume} = 583.1 \times 8 \approx 4664.8 , \text{cm}^3 \]
Conclusion
Thus, the volume of the box (hexagonal prism) is approximately:
\[ \boxed{4664.8} , \text{cm}^3 \]