To find the volume of a regular hexagonal prism, we can use the formula:
\[ V = \text{Base Area} \times \text{Height} \]
The base 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.
Given dimensions:
- Height \(h = 22 , \text{cm}\)
- Length of a side \(s = 8 , \text{cm}\)
Step 1: Calculate the area of the base (regular hexagon)
Using \(s = 8\) cm:
\[ A = \frac{3\sqrt{3}}{2} (8)^2 \] \[ = \frac{3\sqrt{3}}{2} \times 64 \] \[ = 96\sqrt{3} , \text{cm}^2 \]
Step 2: Calculate the volume of the prism
Using height \(h = 22\) cm:
\[ V = A \times h = 96\sqrt{3} \times 22 \] \[ = 2112\sqrt{3} , \text{cm}^3 \]
Step 3: Calculate the numerical value
To find \(V\), we can calculate \( \sqrt{3} \approx 1.732\):
\[ V \approx 2112 \times 1.732 \approx 3660.864 , \text{cm}^3 \]
Step 4: Round to the nearest whole number
\[ \approx 3661 , \text{cm}^3 \]
Conclusion
The correct volume of the regular hexagonal prism is approximately 3661 cm³.
Possible Error Made by the Student
If the student said the volume is 1,619 cm³, they most likely made an error in either calculating the base area of the hexagon or mistakenly used the wrong height. They may have used incorrect values for side length or height, or improperly calculated the area of the hexagon or the volume. They might also have mixed up the units or made a numerical error in multiplication.