In this article, we will walk you through the steps to determine the surface area of a triangular prism. The triangular prism has two triangular bases and three rectangular faces. We will use algebra and geometry to find the surface area based on the given side lengths of the triangle and the height (or depth) of the prism.
Step 1: Show the Surface Area Formula
The surface area (SA) of a triangular prism can be found using the following formula:
\[ SA = 2 \cdot \sqrt{s \cdot (s - a) \cdot (s - b) \cdot (s - c)} + d \cdot (a + b + c) \]
Where:
- \(a, b, c\) are the side lengths of the triangular base.
- \(d\) is the height (or depth) of the prism.
- \(s\) is the semi-perimeter of the triangular base, calculated as:
\[ s = \frac{a + b + c}{2} \]
Step 2: Explain the Formula
- The term \(2 \cdot \sqrt{s \cdot (s - a) \cdot (s - b) \cdot (s - c)}\) represents the area of the two triangular bases.
- The term \(d \cdot (a + b + c)\) 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 the following dimensions:
- Side lengths of the triangular base: \(a = 3\) units, \(b = 4\) units, \(c = 5\) units
- Height (or depth) of the prism: \(d = 6\) units
Step 4: Calculate the Final Value
First, calculate the semi-perimeter \(s\):
\[ s = \frac{a + b + c}{2} = \frac{3 + 4 + 5}{2} = 6 \]
Next, calculate the area of the triangular bases using Heron's formula:
\[ \text{Area of the triangular base} = \sqrt{s \cdot (s - a) \cdot (s - b) \cdot (s - c)} \]
\[ = \sqrt{6 \cdot (6 - 3) \cdot (6 - 4) \cdot (6 - 5)} \]
\[ = \sqrt{6 \cdot 3 \cdot 2 \cdot 1} \]
\[ = \sqrt{36} = 6 \, \text{square units} \]
Since there are two triangular bases, their combined area is:
\[ 2 \cdot 6 = 12 \, \text{square units} \]
Next, calculate the area of the three rectangular faces:
\[ \text{Area of the rectangular faces} = d \cdot (a + b + c) \]
\[ = 6 \cdot (3 + 4 + 5) \]
\[ = 6 \cdot 12 = 72 \, \text{square units} \]
Finally, add the areas of the triangular bases and the rectangular faces to find the total surface area:
\[ SA = 12 + 72 = 84 \, \text{square units} \]
Final Value
The surface area of a triangular prism with side lengths of the base 3, 4, and 5 units, and a height (or depth) of 6 units, is 84 square units.
