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An illustration shows a triangular prism. The front triangular face is visible. The right rectangular face is visible. The visible rectangular face has a width of 5 centimeters and length of 11 centimeters. The sides and edges that are not visible are indicated by dashed lines. The legs of the triangular face are labeled 3 centimeters and 4 centimeters. The hypotenuse shares the same label as the width of the visible rectangular face.

What is the surface area of the triangular prism?
?square centimeters

1 answer

To find the surface area of a triangular prism, we need to calculate the area of each face of the prism.

  1. Calculate the area of the triangular face: The triangular face is a right triangle with legs of lengths 3 cm and 4 cm. The area \( A \) of a triangle is given by the formula:

    \[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

    Here, we can take the base as 3 cm and the height as 4 cm:

    \[ A = \frac{1}{2} \times 3 , \text{cm} \times 4 , \text{cm} = \frac{1}{2} \times 12 , \text{cm}^2 = 6 , \text{cm}^2 \]

    Since there are two identical triangular faces in the prism, the total area of the triangular faces is:

    \[ \text{Total area of triangular faces} = 2 \times 6 , \text{cm}^2 = 12 , \text{cm}^2 \]

  2. Calculate the area of the rectangular faces: The prism has three rectangular faces:

    • One face with dimensions 5 cm (width) and 11 cm (length).
    • The other two faces are the sides that correspond to the edges of the triangle.

    For the face with dimensions 5 cm and 11 cm:

    \[ \text{Area} = \text{width} \times \text{length} = 5 , \text{cm} \times 11 , \text{cm} = 55 , \text{cm}^2 \]

    Next, we need to calculate the areas of the other two rectangular faces:

    • One face corresponds to the leg of 3 cm and the length of the prism (which is the same as the height of the triangular face, 11 cm).
    • The other face corresponds to the leg of 4 cm and the length of the prism (11 cm).

    For the 3 cm face:

    \[ \text{Area} = 3 , \text{cm} \times 11 , \text{cm} = 33 , \text{cm}^2 \]

    For the 4 cm face:

    \[ \text{Area} = 4 , \text{cm} \times 11 , \text{cm} = 44 , \text{cm}^2 \]

  3. Total surface area of the prism: Now, we sum up the areas of all the faces:

    \[ \text{Total Surface Area} = \text{Area of triangular faces} + \text{Area of right face (5 x 11)} + \text{Area of 3 cm face} + \text{Area of 4 cm face} \]

    Substituting the calculated values:

    \[ \text{Total Surface Area} = 12 , \text{cm}^2 + 55 , \text{cm}^2 + 33 , \text{cm}^2 + 44 , \text{cm}^2 \]

    \[ = 12 + 55 + 33 + 44 = 144 , \text{cm}^2 \]

Thus, the surface area of the triangular prism is:

\[ \boxed{144} , \text{square centimeters} \]