In this article, we will guide you through the process of calculating the surface area of a pyramid with an equilateral triangular base. The surface area includes the area of the triangular base and the area of the three triangular faces.
Step-by-Step Guide
Step 1: Show the Surface Area Formula
The surface area (SA) of a pyramid with an equilateral triangular base can be found using the following formula:
\[ SA = \frac{a}{2} \cdot \left(\sqrt{a^2 - \left(\frac{a}{2}\right)^2} + 3 \cdot h_s \right) \]
Where:
- \( a \) is the length of a side of the equilateral triangular base.
- \( h_s \) is the slant height of the pyramid from the midpoint of a side of the base to the apex.
Step 2: Explain the Formula
- The term \( \frac{a}{2} \cdot \sqrt{a^2 - \left(\frac{a}{2}\right)^2} \) represents the area of the equilateral triangular base.
- The term \( 3 \cdot h_s \) represents the combined area of the three triangular faces.
Step 3: Insert Numbers as an Example
Let's consider a pyramid with an equilateral triangular base where:
- Side length of the base: \( a = 6 \) units
- Slant height of the pyramid: \( h_s = 8 \) units
Step 4: Calculate the Final Value
First, calculate the area of the equilateral triangular base:
\[ \text{Area of the triangular base} = \frac{a}{2} \cdot \sqrt{a^2 - \left(\frac{a}{2}\right)^2} \]
\[ = \frac{6}{2} \cdot \sqrt{6^2 - \left(\frac{6}{2}\right)^2} \]
\[ = 3 \cdot \sqrt{36 - 9} \]
\[ = 3 \cdot \sqrt{27} \]
\[ = 3 \cdot 3\sqrt{3} \]
\[ = 9\sqrt{3} \approx 15.59 \, \text{square units} \]
Next, calculate the area of the three triangular faces:
\[ \text{Area of the triangular faces} = 3 \cdot \frac{1}{2} \cdot a \cdot h_s \]
\[ = 3 \cdot \frac{1}{2} \cdot 6 \cdot 8 \]
\[ = 3 \cdot 24 \]
\[ = 72 \, \text{square units} \]
Finally, add the areas of the triangular base and the triangular faces to find the total surface area:
\[ SA = 15.59 + 72 \]
\[ = 87.59 \, \text{square units} \]
Final Value
The surface area of a pyramid with an equilateral triangular base, where the side length of the base is 6 units and the slant height of the pyramid is 8 units, is approximately 87.59 square units.
