Determining the surface area of a tetrahedron, a polyhedron with four triangular faces, involves using its side length. This guide will show you step-by-step how to find the surface area using a specific formula.
Step 1: Show the Surface Area Formula
The formula for the surface area \(SA\) of a tetrahedron is:
\[ SA = \sqrt{3} \cdot a^2 \]
Where:
- \(a\) is the length of each side of the tetrahedron.
Step 2: Explain the Formula
In this formula:
- \(\sqrt{3} \cdot a^2\) represents the total surface area of the four equilateral triangular faces of the tetrahedron. Each equilateral triangle has an area of \(\frac{\sqrt{3}}{4} a^2\), and multiplying by four gives the total surface area.
Step 3: Insert Numbers as an Example
Let's consider a tetrahedron with a side length \(a = 5\) units.
Step 4: Calculate the Final Value
First, we substitute the value into the formula:
\[ SA = \sqrt{3} \cdot 5^2 \]
Next, we calculate the square of the side length:
\[ SA = \sqrt{3} \cdot 25 \]
For \(\sqrt{3} \approx 1.732\):
\[ SA \approx 1.732 \cdot 25 \]
Now, multiply the numbers:
\[ SA \approx 43.3 \, \text{square units} \]
Final Value
The surface area of a tetrahedron with a side length of 5 units is approximately 43.3 square units.
