To calculate the total surface area of a triangular prism, we need to find the area of the triangular faces and the rectangular faces. Here's how we can break it down step-by-step.
Step 1: Calculate the area of the triangular face
Assuming the triangle is a right triangle:
- Let the base of the triangle be 5 cm and the height be 4 cm (as indicated by the measurements).
The area of one triangular face can be calculated using the formula:
\[ \text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} \]
Substituting the values:
\[ \text{Area of triangle} = \frac{1}{2} \times 5 , \text{cm} \times 4 , \text{cm} = \frac{1}{2} \times 20 , \text{cm}^2 = 10 , \text{cm}^2 \]
Step 2: Calculate the area of the rectangular faces
-
Smallest Rectangular Face:
- Dimensions: 3 cm (height) × 5 cm (width)
\[ \text{Area} = \text{height} \times \text{width} = 3 , \text{cm} \times 5 , \text{cm} = 15 , \text{cm}^2 \]
-
Middle Rectangular Face:
- Dimensions: 3 cm (height) × 11 cm (width)
\[ \text{Area} = 3 , \text{cm} \times 11 , \text{cm} = 33 , \text{cm}^2 \]
-
Largest Rectangular Face:
- Dimensions: 4 cm (height) × 11 cm (width)
\[ \text{Area} = 4 , \text{cm} \times 11 , \text{cm} = 44 , \text{cm}^2 \]
Step 3: Total Surface Area of the Triangular Prism
The prism has two triangular faces and three rectangular faces. Since we have already calculated the area of the triangular face as 10 cm², we can now calculate the total surface area.
\[ \text{Total Surface Area} = 2 \times \text{Area of triangular face} + \text{Area of smallest rectangular face} + \text{Area of middle rectangular face} + \text{Area of largest rectangular face} \]
Calculating it step-by-step:
\[ \text{Total Surface Area} = 2 \times 10 , \text{cm}^2 + 15 , \text{cm}^2 + 33 , \text{cm}^2 + 44 , \text{cm}^2 \]
Calculating:
\[ = 20 , \text{cm}^2 + 15 , \text{cm}^2 + 33 , \text{cm}^2 + 44 , \text{cm}^2 = 20 + 15 + 33 + 44 = 112 , \text{cm}^2 \]
Summary of Areas
- Area of ONE triangular face = 10 cm²
- Area of smallest rectangular face = 15 cm²
- Area of middle rectangular face = 33 cm²
- Area of largest rectangular face = 44 cm²
- TOTAL Surface Area of the Triangular Prism = 112 cm²
Make sure to adjust the dimensions used based on the specific identifiers if they're different from what was assumed here.