Finding the surface area of a hexagonal prism involves understanding its geometric properties and applying the correct 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 Hexagonal Prism
The surface area (\( SA \)) of a hexagonal prism can be calculated using the following formula:
\[ SA = 6 \cdot a \cdot h + 3 \cdot \sqrt{3} \cdot a^2 \]
Where:
- \( SA \) is the surface area of the hexagonal prism.
- \( a \) is the side length of the hexagonal base.
- \( h \) is the height of the prism.
Explanation of the Formula
The formula consists of two parts:
1. \( 6 \cdot a \cdot h \): This part calculates the surface area of the six rectangular faces of the prism. Each rectangle has an area of \( a \cdot h \), and there are six such rectangles.
2. \( 3 \cdot \sqrt{3} \cdot a^2 \): This part calculates the combined area of the two hexagonal bases. The term \( 3 \cdot \sqrt{3} \cdot a^2 \) is the area of a regular hexagon.
Example Calculation
Let's go through an example to illustrate how to use this formula.
Given:
- \( a = 4 \) units (the side length of the hexagonal base)
- \( h = 10 \) units (the height of the prism)
We want to find the surface area of the hexagonal prism.
Step-by-Step Calculation
Step 1: Identify the Given Values
Given:
- \( a = 4 \) units
- \( h = 10 \) units
Step 2: Use the Surface Area Formula
\[ SA = 6 \cdot a \cdot h + 3 \cdot \sqrt{3} \cdot a^2 \]
Step 3: Substitute the Given Values into the Formula
\[ SA = 6 \cdot 4 \cdot 10 + 3 \cdot \sqrt{3} \cdot 4^2 \]
Step 4: Calculate the Area of the Rectangular Faces
\[ 6 \cdot 4 \cdot 10 = 240 \]
Step 5: Calculate the Area of the Hexagonal Bases
\[ 3 \cdot \sqrt{3} \cdot 4^2 = 3 \cdot \sqrt{3} \cdot 16 \]
First, compute the constant factor:
\[ 3 \cdot \sqrt{3} \approx 3 \cdot 1.732 \approx 5.196 \]
Then:
\[ 5.196 \cdot 16 = 83.136 \]
Step 6: Sum the Two Parts to Find the Total Surface Area
\[ SA = 240 + 83.136 = 323.136 \]
Final Value
The surface area of a hexagonal prism with a side length of 4 units and a height of 10 units is approximately \( 323.136 \) square units.
