Calculating the surface area of a pentagonal pyramid involves finding the area of the pentagonal base and the area of the five 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 pentagonal pyramid can be calculated using the following formula:
\[ SA = \frac{5}{4} \cdot \tan(54^\circ) \cdot a^2 + 5 \cdot \frac{a}{2} \cdot \sqrt{h^2 + \left( \frac{a \cdot \tan(54^\circ)}{2} \right)^2} \]
Where:
- \( a \) is the side length of the pentagonal base.
- \( h \) is the slant height of the pyramid.
- \( \tan(54^\circ) \) is the tangent of 54 degrees, a constant value related to the internal angles of a pentagon.
Step 2: Explain the Formula
- The term \( \frac{5}{4} \cdot \tan(54^\circ) \cdot a^2 \) represents the area of the pentagonal base.
- The term \( 5 \cdot \frac{a}{2} \cdot \sqrt{h^2 + \left( \frac{a \cdot \tan(54^\circ)}{2} \right)^2} \) represents the combined area of the five triangular faces.
Step 3: Insert Numbers as an Example
Let's consider a pentagonal pyramid where the side length \( a \) of the pentagonal base is 6 units and the slant height \( h \) is 10 units.
Step 4: Calculate the Final Value
First, we need the value of \( \tan(54^\circ) \). Using a calculator, we find:
\[ \tan(54^\circ) \approx 1.376 \]
Now, substitute the given values into the formula:
\[ SA = \frac{5}{4} \cdot 1.376 \cdot 6^2 + 5 \cdot \frac{6}{2} \cdot \sqrt{10^2 + \left( \frac{6 \cdot 1.376}{2} \right)^2} \]
Calculate each part separately:
1. Area of the pentagonal base:
\[ \frac{5}{4} \cdot 1.376 \cdot 6^2 = \frac{5}{4} \cdot 1.376 \cdot 36 = \frac{5}{4} \cdot 49.536 = 61.92 \, \text{square units} \]
2. Area of the triangular faces:
\[ \frac{6 \cdot 1.376}{2} = 4.128 \]
\[ h^2 + 4.128^2 = 10^2 + 4.128^2 = 100 + 17.042 = 117.042 \]
\[ \sqrt{117.042} \approx 10.82 \]
\[ 5 \cdot \frac{6}{2} \cdot 10.82 = 5 \cdot 3 \cdot 10.82 = 162.3 \, \text{square units} \]
Add both parts to find the total surface area:
\[ SA = 61.92 + 162.3 = 224.22 \, \text{square units} \]
Final Value
The surface area of a pentagonal pyramid with a side length of 6 units and a slant height of 10 units is approximately 224.22 square units.
