Finding the surface area of an octahedron, a polyhedron with eight equilateral triangular 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 an octahedron is:
\[ SA = 2 \cdot \sqrt{3} \cdot a^2 \]
Where:
- \(a\) is the length of each side of the octahedron.
Step 2: Explain the Formula
In this formula:
- \(2 \cdot \sqrt{3} \cdot a^2\) represents the total surface area of the eight equilateral triangular faces of the octahedron. Each equilateral triangle has an area of \(\frac{\sqrt{3}}{4} a^2\), and multiplying by eight gives the total surface area.
Step 3: Insert Numbers as an Example
Let's consider an octahedron with a side length \(a = 3\) units.
Step 4: Calculate the Final Value
First, we substitute the value into the formula:
\[ SA = 2 \cdot \sqrt{3} \cdot 3^2 \]
Next, we calculate the square of the side length:
\[ SA = 2 \cdot \sqrt{3} \cdot 9 \]
For \(\sqrt{3} \approx 1.732\):
\[ SA \approx 2 \cdot 1.732 \cdot 9 \]
Now, multiply the numbers:
\[ SA \approx 2 \cdot 15.588 \]
\[ SA \approx 31.176 \, \text{square units} \]
Final Value
The surface area of an octahedron with a side length of 3 units is approximately 31.176 square units.
