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Question

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?

(1 point)
square centimeters
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Answer 1

To find the surface area of the triangular prism, we first need to determine the surface areas of the triangular base and the three rectangular faces.

Step 1: Calculate the area of the triangular base.

The triangular face is a right triangle with:

• One leg = 3 cm
• Other leg = 4 cm

The area of a triangle is given by the formula: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] In this case, we can take the legs as the base and height: \[ \text{Area}_{\triangle} = \frac{1}{2} \times 3 \times 4 = \frac{12}{2} = 6 \text{ cm}^2 \]

Step 2: Determine the dimensions of the prism.

The prism has:

• Width (one side of the triangular face) = 5 cm (this is the height of the prism)
• The length of the prism = 11 cm

Step 3: Calculate the areas of the rectangular faces.

1. Rectangular face corresponding to the triangular base (area on the left):

   • Width = 5 cm (height of the prism),
   • Length = hypotenuse of the triangle.

   To find the hypotenuse (c) of the triangle: \[ c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ cm} \]

   Area of this rectangular face: \[ \text{Area}_{\text{rect}} = \text{width} \times \text{length} = 5 \times 11 = 55 \text{ cm}^2 \]

2. For the right rectangular face (shown in the illustration):

   • Width = 5 cm,
   • Length = length of the prism = 11 cm.

   Area: \[ \text{Area}_{\text{rect}} = 5 \times 11 = 55 \text{ cm}^2 \]

3. For the other rectangular face (side face):

   • Corresponding to the other leg of the triangle.
   • Length = 11 cm.
   • Width = 4 cm (one leg of the triangular base).

   Area: \[ \text{Area}_{\text{rect}} = 4 \times 11 = 44 \text{ cm}^2 \]

Step 4: Total surface area of the prism.

Surface area (SA) of the prism includes:

• 2 triangular bases: \[ \text{Total Area}_{\triangle} = 2 \times 6 = 12 \text{ cm}^2 \]

• 3 rectangular faces: \[ \text{Total Area}_{\text{rect}} = 55 + 55 + 44 = 154 \text{ cm}^2 \]

Final Calculation:

\[ \text{Total Surface Area} = \text{Total Area}{\triangle} + \text{Total Area}{\text{rect}} = 12 + 154 = 166 \text{ cm}^2 \]

Thus, the surface area of the triangular prism is 166 square centimeters.