To find the surface area of the triangular prism, we first need to calculate the area of the triangular face and the areas of the rectangular faces.
Step 1: Area of the Triangular Face
The triangular face is a right triangle with legs of 3 cm and 4 cm. The area \( A \) of a triangle is given by the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] In this case, we can take the legs of the triangle as the base and height: \[ A = \frac{1}{2} \times 3 \text{ cm} \times 4 \text{ cm} = \frac{1}{2} \times 12 \text{ cm}^2 = 6 \text{ cm}^2 \]
Step 2: Area of the Rectangular Faces
The prism has three rectangular faces:
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The visible rectangular face
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The opposite rectangular face (not visible)
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The rectangular face that connects the two triangular faces.
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The visible rectangular face has a width of 5 cm and a length of 11 cm: \[ \text{Area}_{\text{visible}} = \text{width} \times \text{length} = 5 \text{ cm} \times 11 \text{ cm} = 55 \text{ cm}^2 \]
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The opposite rectangular face will have the same dimensions, so: \[ \text{Area}_{\text{opposite}} = 55 \text{ cm}^2 \]
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The connecting rectangular face has a width equal to the hypotenuse of the triangle. To find the hypotenuse \( c \), we can use the Pythagorean theorem: \[ c = \sqrt{(3 \text{ cm})^2 + (4 \text{ cm})^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ cm} \] Now this rectangular face will have a width of 5 cm and a length equal to the length of the prism (which is given as 11 cm): \[ \text{Area}_{\text{connecting}} = \text{width} \times \text{length} = 5 \text{ cm} \times 11 \text{ cm} = 55 \text{ cm}^2 \]
Step 3: Total Surface Area
Now we sum the areas of all the faces:
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Area of two triangular faces (since there are two identical triangular faces): \[ 2 \times 6 \text{ cm}^2 = 12 \text{ cm}^2 \]
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Area of the three rectangular faces: \[ \text{Area}{\text{visible}} + \text{Area}{\text{opposite}} + \text{Area}_{\text{connecting}} = 55 \text{ cm}^2 + 55 \text{ cm}^2 + 55 \text{ cm}^2 = 165 \text{ cm}^2 \]
Final Calculation of Total Surface Area
\[ \text{Total Surface Area} = \text{Area}{\text{triangles}} + \text{Area}{\text{rectangles}} = 12 \text{ cm}^2 + 165 \text{ cm}^2 = 177 \text{ cm}^2 \]
Conclusion
The total surface area of the triangular prism is: \[ \boxed{177} \text{ square centimeters} \]