In this article, we will guide you through the process of calculating the surface area of a triangular prism with equilateral triangular bases. The surface area includes the areas of the two triangular bases and the three rectangular faces.
Step 1: Show the Surface Area Formula
The surface area (SA) of a triangular prism with equilateral triangular bases can be found using the following formula:
\[ SA = 2 \cdot \sqrt{s \cdot (s - a)^3} + d \cdot (3 \cdot a) \]
Where:
- \(a\) is the side length of the equilateral triangular base.
- \(d\) is the height (or depth) of the prism.
- \(s\) is the semi-perimeter of the triangular base, calculated as:
\[ s = \frac{3 \cdot a}{2} \]
Step 2: Explain the Formula
- The term \(2 \cdot \sqrt{s \cdot (s - a)^3}\) represents the area of the two equilateral triangular bases.
- The term \(d \cdot (3 \cdot a)\) represents the combined area of the three rectangular faces of the prism.
Step 3: Insert Numbers as an Example
Let's consider a triangular prism with equilateral triangular bases where:
- Side length of the triangular base: \(a = 4\) units
- Height (or depth) of the prism: \(d = 6\) units
Step 4: Calculate the Final Value
First, calculate the semi-perimeter \(s\):
\[ s = \frac{3 \cdot a}{2} = \frac{3 \cdot 4}{2} = 6 \]
Next, calculate the area of the triangular bases:
\[ \text{Area of one triangular base} = \sqrt{s \cdot (s - a)^3} \]
\[ = \sqrt{6 \cdot (6 - 4)^3} \]
\[ = \sqrt{6 \cdot 2^3} \]
\[ = \sqrt{6 \cdot 8} \]
\[ = \sqrt{48} \approx 6.93 \, \text{square units} \]
Since there are two triangular bases, their combined area is:
\[ 2 \cdot 6.93 \approx 13.86 \, \text{square units} \]
Next, calculate the area of the three rectangular faces:
\[ \text{Area of the rectangular faces} = d \cdot (3 \cdot a) \]
\[ = 6 \cdot (3 \cdot 4) \]
\[ = 6 \cdot 12 = 72 \, \text{square units} \]
Finally, add the areas of the triangular bases and the rectangular faces to find the total surface area:
\[ SA = 13.86 + 72 \approx 85.86 \, \text{square units} \]
Final Value
The surface area of a triangular prism with equilateral triangular bases, where the side length of the base is 4 units and the height (or depth) of the prism is 6 units, is approximately 85.86 square units.
