Calculating the volume of a hexagonal prism is a straightforward process when you use the right formula. This guide will walk you through the steps, explain the necessary formulas, and provide a practical example to ensure you can find the volume of any hexagonal prism.
Volume of a Hexagonal Prism Formula
The volume (\( V \)) of a hexagonal prism can be determined using the formula:
\[ V = \dfrac{3 \cdot \sqrt{3}}{2} \cdot a^2 \cdot h \]
Where:
- \( a \) is the length of one side of the hexagonal base.
- \( h \) is the height of the prism.
Explanation of the Formula
- The term \(\dfrac{3 \cdot \sqrt{3}}{2}\) is derived from the formula for the area of a regular hexagon.
- \( a^2 \) represents the square of the side length of the hexagon.
- Multiplying the area of the hexagonal base by the height (\( h \)) gives the total volume of the prism.
Step-by-Step Calculation
Let’s go through an example to demonstrate how to use this formula to find the volume of a hexagonal prism.
Example: Calculating the Volume of a Hexagonal Prism
1. Identify the given values:
- Side length of the hexagonal base (\( a \)) = 5 units
- Height (\( h \)) = 12 units
2. Substitute the values into the volume formula:
\[ V = \dfrac{3 \cdot \sqrt{3}}{2} \cdot 5^2 \cdot 12 \]
3. Simplify the expression inside the formula:
\[ a^2 = 5^2 = 25 \]
\[ \dfrac{3 \cdot \sqrt{3}}{2} \approx \dfrac{3 \cdot 1.732}{2} \approx \dfrac{5.196}{2} \approx 2.598 \]
4. Substitute back into the formula:
\[ V = 2.598 \cdot 25 \cdot 12 \]
5. Complete the multiplication:
\[ V = 2.598 \cdot 300 \]
\[ V \approx 779.4 \text{ cubic units} \]
Final Volume
The volume of the hexagonal prism with a side length of 5 units and a height of 12 units is approximately 779.4 cubic units.
