Calculating the volume of a hexagonal pyramid involves a specific formula that utilizes the side length of the hexagonal base and the height of the pyramid. This article will guide you through the necessary steps to find the volume, including an example calculation to illustrate the process.
Volume of a Hexagonal Pyramid Formula
To calculate the volume (\( V \)) of a hexagonal pyramid, you can use the following formula:
\[ V = \dfrac{\sqrt{3}}{2} \cdot a^2 \cdot h \]
Where:
- \( a \) is the side length of the hexagonal base.
- \( h \) is the height of the pyramid.
Explanation of the Formula
- The term \( \dfrac{\sqrt{3}}{2} \) is a constant that arises from the geometry of a regular hexagon.
- \( a^2 \) represents the area component of the hexagonal base.
- \( h \) represents the height of the pyramid from the base to the apex.
Step-by-Step Calculation
Let's go through an example to demonstrate how to use this formula.
Example: Calculating the Volume of a Hexagonal Pyramid
1. Identify the given values:
- Side length of the hexagonal base (\( a \)) = 4 units
- Height of the pyramid (\( h \)) = 9 units
2. Substitute the values into the volume formula:
\[ V = \dfrac{\sqrt{3}}{2} \cdot 4^2 \cdot 9 \]
3. Calculate the square of the side length:
\[ 4^2 = 16 \]
4. Substitute and simplify:
\[ V = \dfrac{\sqrt{3}}{2} \cdot 16 \cdot 9 \]
5. Multiply the terms:
\[ V = \dfrac{\sqrt{3}}{2} \cdot 144 \]
6. Calculate the final value:
\[ V = 72 \cdot \sqrt{3} \]
\[ V \approx 72 \cdot 1.732 \]
\[ V \approx 124.704 \]
Final Volume
The volume of the hexagonal pyramid with a side length of 4 units and a height of 9 units is approximately 124.704 cubic units.
