To find the total surface area of the triangular prism, we need to calculate the areas of its components: the triangular bases and the rectangular faces.
Step 1: Area of the Triangular Face
The base of the triangular prism is a right triangle with the following sides: 6 cm, 8 cm, and 10 cm (since this forms a right triangle with the given sides).
To find the area of the triangular face, we can use the formula for the area of a triangle: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] In our case, we can consider 6 cm as the base and 8 cm as the height, so: \[ \text{Area (Triangle)} = \frac{1}{2} \times 6 , \text{cm} \times 8 , \text{cm} = \frac{1}{2} \times 48 , \text{cm}^2 = 24 , \text{cm}^2 \]
Step 2: Area of the Rectangular Faces
The prism has three rectangular faces corresponding to the three sides of the triangular base.
-
Smallest rectangular face (adjacent to the 6 cm base): \[ \text{Area} = \text{base} \times \text{height} = 6 , \text{cm} \times 10 , \text{cm} = 60 , \text{cm}^2 \]
-
Middle rectangular face (adjacent to the 8 cm base): \[ \text{Area} = \text{base} \times \text{height} = 8 , \text{cm} \times 10 , \text{cm} = 80 , \text{cm}^2 \]
-
Largest rectangular face (adjacent to the 9 cm base): \[ \text{Area} = \text{base} \times \text{height} = 9 , \text{cm} \times 10 , \text{cm} = 90 , \text{cm}^2 \]
Step 3: Calculate Total Surface Area
The total surface area of the prism can be calculated as follows: \[ \text{Total Surface Area} = 2 \times \text{Area of triangular face} + \text{Area of rectangular faces} \] \[ \text{Total Surface Area} = 2 \times 24 , \text{cm}^2 + 60 , \text{cm}^2 + 80 , \text{cm}^2 + 90 , \text{cm}^2 \] \[ = 48 , \text{cm}^2 + 60 , \text{cm}^2 + 80 , \text{cm}^2 + 90 , \text{cm}^2 \] \[ = 48 + 60 + 80 + 90 = 278 , \text{cm}^2 \]
Final Answers
- Area of ONE triangular face = 24 cm²
- Area of the smallest rectangular face = 60 cm²
- Area of the middle rectangular face = 80 cm²
- Area of the largest rectangular face = 90 cm²
- TOTAL Surface Area of the Triangular Prism = 278 cm²