Question

Show your steps and explain how to find the surface area of the triangular prism below.
it has a hight of 6
a base of 19
a length of 8
and a back of 10

Answer 1

To find the surface area of a triangular prism, we need to calculate the areas of the two triangular bases and the three rectangular sides. The surface area (SA) is given by the formula:

\[
SA = 2 \times \text{Area of triangular base} + \text{Area of rectangular sides}
\]

### Step 1: Determine the area of the triangular base

The area of a triangle can be calculated using the formula:

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

From the information given:
- Base = 19
- Height = 6

Plugging in these values:

\[
\text{Area} = \frac{1}{2} \times 19 \times 6 = \frac{1}{2} \times 114 = 57
\]

### Step 2: Calculate the area of the triangular bases

Since there are two triangular bases:

\[
\text{Area of both triangular bases} = 2 \times 57 = 114
\]

### Step 3: Calculate the areas of the rectangular sides

A triangular prism has three rectangular sides, and their areas can be calculated as follows:

1. **Rectangle 1** (the side corresponding to the base of the triangle):
- Width (base of the triangle) = 19
- Length of the prism = 8
- Area = Width × Length = \(19 \times 8 = 152\)

2. **Rectangle 2** (the side corresponding to the height of the triangle):
- Width (height of the triangle) = 6
- Length of the prism = 8
- Area = Width × Length = \(6 \times 8 = 48\)

3. **Rectangle 3** (the side corresponding to the back of the triangle):
- Width (back of the triangle) = 10
- Length of the prism = 8
- Area = Width × Length = \(10 \times 8 = 80\)

### Step 4: Sum the areas of the rectangular sides

Now we add the areas of the three rectangular sides:

\[
\text{Area of rectangular sides} = 152 + 48 + 80 = 280
\]

### Step 5: Calculate the total surface area

Now we combine the area of the triangular bases and the area of the rectangular sides:

\[
SA = \text{Area of both triangular bases} + \text{Area of rectangular sides} = 114 + 280 = 394
\]

### Conclusion

The surface area of the triangular prism is:

\[
\boxed{394}
\]

This is the final answer for the surface area of the given triangular prism.