To calculate the surface area of a triangular prism, we need to determine the areas of the two triangular bases and the three rectangular sides. Given the dimensions:
- W (width of the base triangle) = 27.9 cm
- L (length of the prism) = 30.5 cm
- H (height of the base triangle) = 5.1 cm
- s (slant height) = 28.4 cm
Step 1: Area of the Triangular Base
The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
For our triangular base:
\[ A = \frac{1}{2} \times 27.9 \times 5.1 = \frac{1}{2} \times 142.29 = 71.145 , \text{cm}^2 \]
Step 2: Total Area of Two Triangular Bases
Since there are two identical triangular bases:
\[ \text{Area of two bases} = 2A = 2 \times 71.145 = 142.29 , \text{cm}^2 \]
Step 3: Area of the Rectangular Sides
Now we calculate the areas of the three rectangular sides:
-
Base Rectangle (length × width): \[ \text{Area} = L \times W = 30.5 \times 27.9 = 852.45 , \text{cm}^2 \]
-
Side Rectangle 1 (length × height): \[ \text{Area} = L \times H = 30.5 \times 5.1 = 155.55 , \text{cm}^2 \]
-
Side Rectangle 2 (length × slant height): \[ \text{Area} = L \times s = 30.5 \times 28.4 = 865.42 , \text{cm}^2 \]
Step 4: Total Area of the Rectangular Sides
Combining the areas of the three rectangular sides:
\[ \text{Total Area of Rectangles} = 852.45 + 155.55 + 865.42 = 1873.42 , \text{cm}^2 \]
Step 5: Total Surface Area
Now we can find the total surface area \( S.A. \):
\[ S.A. = \text{Area of two bases} + \text{Total Area of Rectangles} \] \[ S.A. = 142.29 + 1873.42 = 2015.71 , \text{cm}^2 \]
Final Step: Rounding
Rounding to the nearest whole number:
\[ \text{S.A.} \approx 2016 , \text{cm}^2 \]
Therefore, the amount of wrapping paper needed to wrap the toy is approximately 2016 cm².