Calculating the surface area of a hexagonal pyramid involves finding the area of the hexagonal base and the area of the six triangular faces. This article will guide you through the steps using a specific formula.
Step-by-Step Guide
Step 1: Show the Surface Area Formula
The surface area (SA) of a hexagonal pyramid can be calculated using the following formula:
\[ SA = \frac{3 \cdot \sqrt{3}}{2} \cdot a^2 + 3 \cdot a \cdot \sqrt{h^2 + \frac{3 \cdot a^2}{4}} \]
Where:
- \( a \) is the side length of the hexagonal base.
- \( h \) is the slant height of the pyramid.
- \( \frac{3 \cdot \sqrt{3}}{2} \cdot a^2 \) is the area of the hexagonal base.
- \( 3 \cdot a \cdot \sqrt{h^2 + \frac{3 \cdot a^2}{4}} \) is the combined area of the six triangular faces.
Step 2: Explain the Formula
- The term \( \frac{3 \cdot \sqrt{3}}{2} \cdot a^2 \) represents the area of the hexagonal base.
- The term \( 3 \cdot a \cdot \sqrt{h^2 + \frac{3 \cdot a^2}{4}} \) represents the combined area of the six triangular faces.
Step 3: Insert Numbers as an Example
Let's consider a hexagonal pyramid where the side length \( a \) of the hexagonal base is 4 units and the slant height \( h \) is 10 units.
Step 4: Calculate the Final Value
First, substitute the given values into the formula:
\[ SA = \frac{3 \cdot \sqrt{3}}{2} \cdot 4^2 + 3 \cdot 4 \cdot \sqrt{10^2 + \frac{3 \cdot 4^2}{4}} \]
Calculate each part separately:
1. Area of the hexagonal base:
\[ \frac{3 \cdot \sqrt{3}}{2} \cdot 4^2 = \frac{3 \cdot \sqrt{3}}{2} \cdot 16 = 24 \cdot \sqrt{3} \approx 24 \cdot 1.732 = 41.57 \, \text{square units} \]
2. Area of the triangular faces:
\[ \frac{3 \cdot 4^2}{4} = \frac{3 \cdot 16}{4} = 12 \]
\[ h^2 + 12 = 10^2 + 12 = 100 + 12 = 112 \]
\[ \sqrt{112} \approx 10.58 \]
\[ 3 \cdot 4 \cdot 10.58 = 3 \cdot 42.32 = 126.96 \, \text{square units} \]
Add both parts to find the total surface area:
\[ SA = 41.57 + 126.96 = 168.53 \, \text{square units} \]
Final Value
The surface area of a hexagonal pyramid with a side length of 4 units and a slant height of 10 units is approximately 168.53 square units.
