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?
?square centimeters

Answer 1

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:

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