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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

Answer 1

To calculate the surface area of the triangular prism, we first need to determine the area of the triangular base and then the area of the rectangular faces.

Step 1: Calculate the area of the triangular base

The triangular face is a right triangle with legs measuring 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 use the legs of the triangle as the base and height. So,

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

Step 2: Calculate the area of the rectangular faces

The prism has three rectangular faces: two of the same size corresponding to the triangular base and one rectangular face given in the problem.

1. Area of the rectangular faces corresponding to the triangle:

   The height of the prism is the same as the hypotenuse of the triangle, which can be calculated using the Pythagorean theorem:

   \[ \text{hypotenuse} = \sqrt{(3 , \text{cm})^2 + (4 , \text{cm})^2} = \sqrt{9 + 16} = \sqrt{25} = 5 , \text{cm} \]

   The two rectangular faces corresponding to the legs of the triangle:

   • For the side of length 3 cm: \[ \text{Area} = \text{leg} \times \text{height} = 3 , \text{cm} \times 11 , \text{cm} = 33 , \text{cm}^2 \]
   • For the side of length 4 cm: \[ \text{Area} = \text{leg} \times \text{height} = 4 , \text{cm} \times 11 , \text{cm} = 44 , \text{cm}^2 \]

2. Area of the base rectangle: The width of this rectangular face is given as 5 cm, and its height is 11 cm. Thus,

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

Step 3: Total surface area calculation

Add all the areas together to find the total surface area of the prism:

\[ \text{Total Surface Area} = \text{Area of triangular base} + 2(\text{Area corresponding to legs}) + \text{Area of base triangle} \] \[ = 6 , \text{cm}^2 + 33 , \text{cm}^2 + 44 , \text{cm}^2 + 55 , \text{cm}^2 \] \[ = 6 + 33 + 44 + 55 = 138 , \text{cm}^2 \]

Thus, the surface area of the triangular prism is:

\[ \boxed{138} \text{ square centimeters} \]