Finding the surface area of a dodecahedron, a polyhedron with twelve regular pentagonal faces, involves using the length of its sides. This guide will provide a step-by-step process to determine the surface area using a specific formula.
Step 1: Show the Surface Area Formula
The formula for the surface area \(SA\) of a dodecahedron is:
\[ SA = 3 \cdot \sqrt{25 + 10 \cdot \sqrt{5}} \cdot a^2 \]
Where:
- \(a\) is the length of each side of the dodecahedron.
Step 2: Explain the Formula
In this formula:
- \(3 \cdot \sqrt{25 + 10 \cdot \sqrt{5}} \cdot a^2\) represents the total surface area of the twelve regular pentagonal faces of the dodecahedron. Each face has an area based on the side length \(a\), and the factor \(3 \cdot \sqrt{25 + 10 \cdot \sqrt{5}}\) accounts for the shape and number of faces.
Step 3: Insert Numbers as an Example
Let's consider a dodecahedron with a side length \(a = 2\) units.
Step 4: Calculate the Final Value
First, we substitute the value into the formula:
\[ SA = 3 \cdot \sqrt{25 + 10 \cdot \sqrt{5}} \cdot 2^2 \]
Next, we calculate the square of the side length:
\[ SA = 3 \cdot \sqrt{25 + 10 \cdot \sqrt{5}} \cdot 4 \]
We simplify the inner expression:
\[ \sqrt{25 + 10 \cdot \sqrt{5}} \]
For \(\sqrt{5} \approx 2.236\):
\[ 10 \cdot \sqrt{5} \approx 10 \cdot 2.236 = 22.36 \]
So,
\[ \sqrt{25 + 22.36} = \sqrt{47.36} \approx 6.88 \]
Now, multiply by the constants:
\[ SA = 3 \cdot 6.88 \cdot 4 \]
\[ SA = 20.64 \cdot 4 \]
\[ SA = 82.56 \, \text{square units} \]
Final Value
The surface area of a dodecahedron with a side length of 2 units is approximately 82.56 square units.
