To find the surface area of a triangular prism, we need to consider two components: the area of the triangular bases and the area of the three rectangular lateral faces.
In this case, the right triangle has the dimensions:
- Base (b) = 6 in.
- Height (h) = 4.5 in.
- Hypotenuse (lever) = 7.5 in.
Step 1: Area of the triangular base
The area \( A \) of the triangular base is given by the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] Substituting the values: \[ A = \frac{1}{2} \times 6 , \text{in} \times 4.5 , \text{in} = \frac{1}{2} \times 27 , \text{in}^2 = 13.5 , \text{in}^2 \]
Step 2: Area of the triangular bases
Since there are two triangular bases: \[ \text{Area of bases} = 2 \times A = 2 \times 13.5 , \text{in}^2 = 27 , \text{in}^2 \]
Step 3: Area of the rectangular lateral faces
The lateral surface area can be calculated using the perimeter of the triangular base and the length of the prism (12 in):
Find the sides of the triangle:
- Base = 6 in
- Height = 4.5 in
- Hypotenuse = 7.5 in (given)
Calculate the perimeter of the triangular base:
\[ \text{Perimeter} = 6 , \text{in} + 4.5 , \text{in} + 7.5 , \text{in} = 18 , \text{in} \]
Calculate the lateral surface area:
The lateral surface area \( L \) is given by: \[ L = \text{Perimeter} \times \text{Length} = 18 , \text{in} \times 12 , \text{in} = 216 , \text{in}^2 \]
Step 4: Total surface area
Finally, the total surface area \( S \) of the prism is: \[ S = \text{Area of bases} + \text{Lateral area} = 27 , \text{in}^2 + 216 , \text{in}^2 = 243, \text{in}^2 \]
Thus, the total surface area of the triangular prism is: \[ \boxed{243 , \text{in}^2} \]