To calculate the surface area of a triangular prism, we can use the formula:
\[ S.A. = \text{Base Area} + \text{Lateral Area} \]
Step 1: Calculate the Base Area
The base area is given by the area of the triangular face of the prism. Since the triangular face is a right triangle, we can use the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
From the information provided:
- Base (b) = 27.9 cm
- Height (h) = 5.1 cm
\[ \text{Area} = \frac{1}{2} \times 27.9 \times 5.1 = \frac{1}{2} \times 142.29 = 71.145 , \text{cm}^2 \]
Step 2: Calculate the Lateral Area
The lateral area consists of the rectangles that connect the triangular bases. There are three rectangles (sides) to consider:
- Rectangle corresponding to the base of the triangle.
- Rectangle corresponding to the height of the triangle (perpendicular side).
- Rectangle corresponding to the hypotenuse of the triangle.
Base rectangle:
- Length = height of prism (30.5 cm), Width = base of triangle (27.9 cm) \[ \text{Area} = 30.5 \times 27.9 = 852.95 , \text{cm}^2 \]
Height rectangle:
- Length = height of prism (30.5 cm), Width = height of triangle (5.1 cm) \[ \text{Area} = 30.5 \times 5.1 = 155.55 , \text{cm}^2 \]
Hypotenuse rectangle:
- Length = height of prism (30.5 cm), Width = length of hypotenuse (28.4 cm) \[ \text{Area} = 30.5 \times 28.4 = 865.42 , \text{cm}^2 \]
Step 3: Add the Areas Together
Now, we sum all the areas calculated:
- Base Area (2 triangular faces): \( 2 \times 71.145 = 142.29 , \text{cm}^2\)
- Lateral Areas (3 rectangles):
\[ \text{Lateral Area} = 852.95 + 155.55 + 865.42 = 1873.92 , \text{cm}^2 \]
Final Step: Total Surface Area
Combining the base area and the lateral area:
\[ S.A. = \text{Base Area} + \text{Lateral Area} = 142.29 + 1873.92 = 2016.21 , \text{cm}^2 \]
Rounding to the nearest whole number:
\[ S.A. \approx 2016 , \text{cm}^2 \]
Final Answer:
S.A. = 2016 cm²