Question

A block of wood has the shape of a triangular prism. The bases are right triangles. Find its surface area.
6 in.
7.5 in.
4.5 in.
12 in.
Question content area bottom
Part 1
The surface area is
  
enter your response here in.squared.

Answer 1

To find the surface area of a triangular prism, we need to consider two components: the area of the triangular bases and the area of the three rectangular lateral faces.

In this case, the right triangle has the dimensions:
- Base (b) = 6 in.
- Height (h) = 4.5 in.
- Hypotenuse (lever) = 7.5 in.

### Step 1: Area of the triangular base
The area \( A \) of the triangular base is given by the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Substituting the values:
\[
A = \frac{1}{2} \times 6 \, \text{in} \times 4.5 \, \text{in} = \frac{1}{2} \times 27 \, \text{in}^2 = 13.5 \, \text{in}^2
\]

### Step 2: Area of the triangular bases
Since there are two triangular bases:
\[
\text{Area of bases} = 2 \times A = 2 \times 13.5 \, \text{in}^2 = 27 \, \text{in}^2
\]

### Step 3: Area of the rectangular lateral faces
The lateral surface area can be calculated using the perimeter of the triangular base and the length of the prism (12 in):

#### Find the sides of the triangle:
1. **Base = 6 in**
2. **Height = 4.5 in**
3. **Hypotenuse = 7.5 in** (given)

#### Calculate the perimeter of the triangular base:
\[
\text{Perimeter} = 6 \, \text{in} + 4.5 \, \text{in} + 7.5 \, \text{in} = 18 \, \text{in}
\]

#### Calculate the lateral surface area:
The lateral surface area \( L \) is given by:
\[
L = \text{Perimeter} \times \text{Length} = 18 \, \text{in} \times 12 \, \text{in} = 216 \, \text{in}^2
\]

### Step 4: Total surface area
Finally, the total surface area \( S \) of the prism is:
\[
S = \text{Area of bases} + \text{Lateral area} = 27 \, \text{in}^2 + 216 \, \text{in}^2 = 243\, \text{in}^2
\]

Thus, the total surface area of the triangular prism is:
\[
\boxed{243 \, \text{in}^2}
\]