Calculating the surface area of a pentagonal prism involves understanding its geometric properties and applying the right formulas. This article will guide you through the process using a straightforward algebraic formula, complete with a detailed example.
Formula to Calculate the Surface Area of a Pentagonal Prism
The surface area (\( SA \)) of a pentagonal prism can be calculated using the following formula:
\[ SA = 5 \cdot a \cdot h + \dfrac{\sqrt{5 \cdot (5 + 2 \cdot \sqrt{5})}}{2} \cdot a^2 \]
Where:
- \( SA \) is the surface area of the pentagonal prism.
- \( a \) is the side length of the pentagonal base.
- \( h \) is the height of the prism.
Explanation of the Formula
The formula consists of two parts:
1. \( 5 \cdot a \cdot h \): This part calculates the surface area of the five rectangular faces of the prism. Each rectangle has an area of \( a \cdot h \), and there are five such rectangles.
2. \( \dfrac{\sqrt{5 \cdot (5 + 2 \cdot \sqrt{5})}}{2} \cdot a^2 \): This part calculates the combined area of the two pentagonal bases. The term \( \dfrac{\sqrt{5 \cdot (5 + 2 \cdot \sqrt{5})}}{2} \cdot a^2 \) is the area of a regular pentagon.
Example Calculation
Let's go through an example to illustrate how to use this formula.
Given:
- \( a = 4 \) units (the side length of the pentagonal base)
- \( h = 10 \) units (the height of the prism)
We want to find the surface area of the pentagonal prism.
Step 1: Identify the Given Values
Given:
- \( a = 4 \) units
- \( h = 10 \) units
Step 2: Use the Surface Area Formula
\[ SA = 5 \cdot a \cdot h + \dfrac{\sqrt{5 \cdot (5 + 2 \cdot \sqrt{5})}}{2} \cdot a^2 \]
Step 3: Substitute the Given Values into the Formula
\[ SA = 5 \cdot 4 \cdot 10 + \dfrac{\sqrt{5 \cdot (5 + 2 \cdot \sqrt{5})}}{2} \cdot 4^2 \]
Step 4: Calculate the Area of the Rectangular Faces
\[ 5 \cdot 4 \cdot 10 = 200 \]
Step 5: Calculate the Area of the Pentagonal Bases
\[ \dfrac{\sqrt{5 \cdot (5 + 2 \cdot \sqrt{5})}}{2} \cdot 4^2 = \dfrac{\sqrt{5 \cdot (5 + 2 \cdot \sqrt{5})}}{2} \cdot 16 \]
First, compute the constant factor:
\[ \sqrt{5 \cdot (5 + 2 \cdot \sqrt{5})} \approx \sqrt{5 \cdot 9.472} \approx \sqrt{47.36} \approx 6.88 \]
Then:
\[ \dfrac{6.88}{2} \cdot 16 = 3.44 \cdot 16 = 55.04 \]
Step 6: Sum the Two Parts to Find the Total Surface Area
\[ SA = 200 + 55.04 = 255.04 \]
Final Value
The surface area of a pentagonal prism with a side length of 4 units and a height of 10 units is approximately \( 255.04 \) square units.
