To find the surface area of a triangular prism, we need to calculate the area of the two triangular bases and the three rectangular faces.
Step 1: Calculate the area of the triangular base
For a right triangle, the area can be calculated using the formula: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] Given that the base is 3 cm and the height is 4 cm: \[ \text{Area} = \frac{1}{2} \times 3 , \text{cm} \times 4 , \text{cm} = \frac{12}{2} = 6 , \text{cm}^2 \]
Step 2: Calculate the area of both triangular bases
Since there are two identical triangular bases, we multiply the area of one triangle by 2: \[ \text{Total area of triangular bases} = 2 \times 6 , \text{cm}^2 = 12 , \text{cm}^2 \]
Step 3: Calculate the area of the rectangular faces
The rectangular faces of the prism correspond to the three sides of the triangular base and the height of the prism:
-
Rectangle corresponding to the base (3 cm): \[ \text{Area} = \text{base} \times \text{height of prism} = 3 , \text{cm} \times 8 , \text{cm} = 24 , \text{cm}^2 \]
-
Rectangle corresponding to the height (4 cm): \[ \text{Area} = \text{height} \times \text{height of prism} = 4 , \text{cm} \times 8 , \text{cm} = 32 , \text{cm}^2 \]
-
Rectangle corresponding to the hypotenuse (5 cm): \[ \text{Area} = \text{hypotenuse} \times \text{height of prism} = 5 , \text{cm} \times 8 , \text{cm} = 40 , \text{cm}^2 \]
Step 4: Calculate the total area of the rectangular faces
Adding the areas of all three rectangular faces: \[ \text{Total area of rectangular faces} = 24 , \text{cm}^2 + 32 , \text{cm}^2 + 40 , \text{cm}^2 = 96 , \text{cm}^2 \]
Step 5: Calculate the total surface area of the prism
Finally, the total surface area of the prism is the sum of the areas of the triangular bases and the rectangular faces: \[ \text{Total surface area} = \text{Total area of triangular bases} + \text{Total area of rectangular faces} = 12 , \text{cm}^2 + 96 , \text{cm}^2 = 108 , \text{cm}^2 \]
Thus, the surface area of the prism is: \[ \boxed{108 , \text{cm}^2} \]