To determine the total surface area of the triangular prism, we'll need to calculate the areas of the triangular face and the three rectangular faces (smallest, middle, and largest).
1. Area of the Triangular Face:
The triangle has sides of \(3 \text{ cm}\), \(4 \text{ cm}\), and \(5 \text{ cm}\). Since this is a right triangle (as \(3^2 + 4^2 = 5^2\)), we can use the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
We can take the base as \(3 \text{ cm}\) and the height as \(4 \text{ cm}\):
\[ \text{Area of triangular face} = \frac{1}{2} \times 3 \times 4 = \frac{12}{2} = 6 \text{ cm}^2 \]
2. Area of the Rectangular Faces:
Next, we calculate the areas of the rectangular faces. The dimensions of the prism are \(6 \text{ cm}\) (the height of the prism) times the lengths of each side of the triangular face.
- Smallest rectangular face (3 cm side):
\[ \text{Area} = \text{length} \times \text{height} = 3 \times 6 = 18 \text{ cm}^2 \]
- Middle rectangular face (4 cm side):
\[ \text{Area} = 4 \times 6 = 24 \text{ cm}^2 \]
- Largest rectangular face (5 cm side):
\[ \text{Area} = 5 \times 6 = 30 \text{ cm}^2 \]
3. Total Surface Area:
The total surface area is the sum of the area of the two triangular faces and the areas of the three rectangular faces. There are two identical triangular faces.
\[ \text{Total surface area} = 2(\text{Area of triangular face}) + \text{Area of smallest rectangular face} + \text{Area of middle rectangular face} + \text{Area of largest rectangular face} \]
Calculating this gives:
\[ \text{Total surface area} = 2 \times 6 + 18 + 24 + 30 = 12 + 18 + 24 + 30 = 84 \text{ cm}^2 \]
Thus, we can summarize:
- Area of ONE triangular face = \(6\) square centimeters
- Area of the smallest rectangular face = \(18\) square centimeters
- Area of the middle rectangular face = \(24\) square centimeters
- Area of the largest rectangular face = \(30\) square centimeters
- Total surface area of the triangular prism = \(84\) square centimeters