To find the surface area of a triangular prism, you need the area of the triangular base and the area of the three rectangular sides.
In this case, you have a triangular prism with:
- Base sides of the triangle: 3 cm, 4 cm, and 5 cm
- Height of the prism: 11 cm
Step 1: Calculate the area of the triangular base.
To find the area of the triangle, we can use Heron's formula. First, we need to find the semi-perimeter \( s \):
\[ s = \frac{a + b + c}{2} = \frac{3 + 4 + 5}{2} = 6 \text{ cm} \]
Now, using Heron's formula for the area \( A \):
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting in the values:
\[ A = \sqrt{6(6-3)(6-4)(6-5)} = \sqrt{6 \cdot 3 \cdot 2 \cdot 1} = \sqrt{36} = 6 \text{ cm}^2 \]
Step 2: Calculate the surface area of the prism.
The surface area \( SA \) of the prism is given by the formula:
\[ SA = 2 \times A + P \times h \]
Where:
- \( A \) is the area of the triangular base (already calculated as 6 cm²),
- \( P \) is the perimeter of the triangular base,
- \( h \) is the height of the prism (11 cm).
Calculate the perimeter \( P \):
\[ P = a + b + c = 3 + 4 + 5 = 12 \text{ cm} \]
Substitute into the surface area formula:
\[ SA = 2 \times 6 + 12 \times 11 \]
Calculating this:
\[ SA = 12 + 132 = 144 \text{ cm}^2 \]
Final Answer:
The surface area of the triangular prism is \( 144 \text{ cm}^2 \).