Determining the surface area of an icosahedron, a polyhedron with 20 equilateral triangular faces, involves using the length of its sides. This guide will walk you through the process step-by-step using the appropriate formula.
Step 1: Show the Surface Area Formula
The formula for the surface area \(SA\) of an icosahedron is:
\[ SA = 5 \cdot \sqrt{3} \cdot a^2 \]
Where:
- \(a\) is the length of each side of the icosahedron.
Step 2: Explain the Formula
In this formula:
- \(5 \cdot \sqrt{3} \cdot a^2\) represents the combined area of the 20 equilateral triangular faces. The factor \(5 \cdot \sqrt{3}\) arises from summing up the areas of all the faces, each of which has an area formula of \(\frac{\sqrt{3}}{4} \cdot a^2\).
Step 3: Insert Numbers as an Example
Let's consider an icosahedron with a side length \(a = 3\) units.
Step 4: Calculate the Final Value
First, substitute the value into the formula:
\[ SA = 5 \cdot \sqrt{3} \cdot 3^2 \]
Next, calculate the square of the side length:
\[ SA = 5 \cdot \sqrt{3} \cdot 9 \]
Then, multiply the constants:
\[ SA = 45 \cdot \sqrt{3} \]
Using the approximate value of \(\sqrt{3} \approx 1.732\):
\[ SA = 45 \cdot 1.732 \]
\[ SA \approx 77.94 \, \text{square units} \]
Final Value
The surface area of an icosahedron with a side length of 3 units is approximately 77.94 square units.
