To find the surface area of a triangular prism, we can use the following formula:
\[ S.A. = \text{Base Area} + \text{Lateral Area} \]
Step 1: Calculate the Area of the Triangular Base
The triangle given is a right triangle. The area \( A \) of a right triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case:
- Base = 27.9 cm
- Height (perpendicular side) = 5.1 cm
\[ A = \frac{1}{2} \times 27.9 \times 5.1 = \frac{1}{2} \times 142.29 \approx 71.145 \ \text{cm}^2 \]
Step 2: Calculate the Lateral Surface Area
The lateral area consists of the three rectangles that wrap around the sides of the triangular prism. The lateral surface area \( L \) can be calculated as follows:
\[ L = \text{Perimeter of the base triangle} \times \text{length} \]
Step 2.1: Calculate the sides of the triangular base
Perimeter includes:
- Base = 27.9 cm
- Height = 5.1 cm
- Hypotenuse = 28.4 cm
\[ \text{Perimeter} = 27.9 + 5.1 + 28.4 = 61.4 \ \text{cm} \]
Step 2.2: Calculate the lateral area
\[ L = \text{Perimeter} \times \text{length} = 61.4 \times 30.5 \]
Calculating:
\[ L = 1873.7 \ \text{cm}^2 \]
Step 3: Calculate Total Surface Area
Combining the area of the triangular base and the lateral area:
Since there are two triangular bases in a prism, we add the area of the triangle twice:
\[ S.A. = 2 \times A + L = 2 \times 71.145 + 1873.7 \]
Calculating:
\[ S.A. = 142.29 + 1873.7 = 2016.19 \ \text{cm}^2 \]
Step 4: Round to the Nearest Whole Number
Rounding \( 2016.19 \) gives us:
\[ S.A. = 2016 \ \text{cm}^2 \]
Final Answer
The amount of wrapping paper needed to wrap the toy is \( \boxed{2016} \ \text{cm}^2 \).