Understanding how to calculate the volume of a pentagonal prism can be useful for various geometric and practical applications. This article will guide you through the process using an algebraic formula. We will break down the formula, explain each component, and provide a step-by-step example calculation.
Volume of a Pentagonal Prism Formula
The volume (\( V \)) of a pentagonal prism can be calculated using the following formula:
\[ V = \dfrac{1}{4} \cdot \sqrt{5 \cdot (5 + 2 \cdot \sqrt{5})} \cdot a^2 \cdot h \]
Where:
- \( a \) is the length of one side of the pentagonal base.
- \( h \) is the height of the prism.
Explanation of the Formula
- The term \(\dfrac{1}{4} \cdot \sqrt{5 \cdot (5 + 2 \cdot \sqrt{5})}\) represents a constant that arises from the geometric properties of a regular pentagon.
- \( a^2 \) is the area of one of the triangular sections of the pentagonal base.
- Multiplying the area of the 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 pentagonal prism.
Example: Calculating the Volume of a Pentagonal Prism
1. Identify the given values:
- Side length of the pentagonal base (\( a \)) = 4 units
- Height (\( h \)) = 10 units
2. Substitute the values into the volume formula:
\[ V = \dfrac{1}{4} \cdot \sqrt{5 \cdot (5 + 2 \cdot \sqrt{5})} \cdot 4^2 \cdot 10 \]
3. Simplify the expression inside the square root:
\[ \sqrt{5 \cdot (5 + 2 \cdot \sqrt{5})} \]
Calculate the inner term:
\[ 5 + 2 \cdot \sqrt{5} \approx 5 + 4.472 \approx 9.472 \]
Now, multiply by 5:
\[ 5 \cdot 9.472 \approx 47.36 \]
Finally, take the square root:
\[ \sqrt{47.36} \approx 6.88 \]
4. Substitute back into the formula:
\[ V = \dfrac{1}{4} \cdot 6.88 \cdot 16 \cdot 10 \]
5. Complete the multiplication:
\[ V = \dfrac{1}{4} \cdot 6.88 \cdot 160 \]
\[ V = 1.72 \cdot 160 \]
\[ V = 275.2 \text{ cubic units} \]
Final Volume
The volume of the pentagonal prism with a side length of 4 units and a height of 10 units is approximately 275.2 cubic units.
