Calculating the volume of a pentagonal pyramid involves a specific formula that incorporates the base's side length, the height of the pyramid, and trigonometric functions. This article will guide you through the necessary steps to find the volume, including an example calculation to illustrate the process.
Volume of a Pentagonal Pyramid Formula
To calculate the volume (\( V \)) of a pentagonal pyramid, you can use the following formula:
\[ V = \dfrac{5}{12} \cdot \tan(54^\circ) \cdot h \cdot a^3 \]
Where:
- \( a \) is the side length of the pentagonal base.
- \( h \) is the height of the pyramid.
Explanation of the Formula
- The term \( \dfrac{5}{12} \) is a constant that scales the volume for a pentagonal base.
- \( \tan(54^\circ) \) involves the tangent of 54 degrees, a key angle in the geometry of a regular pentagon.
- \( h \) represents the height of the pyramid from the base to the apex.
- \( a^3 \) is the cube of the side length, combining the area of the base and the height in three-dimensional space.
Step-by-Step Calculation
Let's go through an example to demonstrate how to use this formula.
Example: Calculating the Volume of a Pentagonal Pyramid
1. Identify the given values:
- Side length of the pentagonal base (\( a \)) = 3 units
- Height of the pyramid (\( h \)) = 6 units
2. Substitute the values into the volume formula:
\[ V = \dfrac{5}{12} \cdot \tan(54^\circ) \cdot 6 \cdot 3^3 \]
3. Calculate the tangent of 54 degrees:
\[ \tan(54^\circ) \approx 1.376 \]
4. Calculate the cube of the side length:
\[ 3^3 = 27 \]
5. Substitute and simplify:
\[ V = \dfrac{5}{12} \cdot 1.376 \cdot 6 \cdot 27 \]
6. Multiply the terms:
\[ V \approx \dfrac{5}{12} \cdot 1.376 \cdot 162 \]
\[ V \approx \dfrac{5}{12} \cdot 222.912 \]
\[ V \approx \dfrac{1114.56}{12} \]
\[ V \approx 92.88 \]
Final Volume
The volume of the pentagonal pyramid with a side length of 3 units and a height of 6 units is approximately 92.88 cubic units.
