Question

Consider the figure below.

10cm

8cm

6cm

9cm



Determine the total surface area of the triangular prism. The formula chart is at the bottom of the page.

(5 points)
Area of each triangular face =
square centimeters

Area of the smallest rectangular face =
square centimeters

Area of the middle rectangular face =
square centimeters

Area of the largest rectangular face =
square centimeters

TOTAL Surface Area of the Triangular Prism =
square centimeters

Answer 1

To find the total surface area of the triangular prism, we need to calculate the area of the triangular faces and the rectangular faces.

### Step 1: Area of each triangular face

For a triangular prism, if we know the lengths of the sides of the triangular base, we can calculate the area using Heron's formula, or we could use the formula for the area of a triangle if we know the base and height.

Here, from the information given, let's assume that the triangular face is a right triangle with the following dimensions:

- Base = 6 cm
- Height = 8 cm

The area of a triangle is calculated using the formula:
\[
\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Substituting the values:
\[
\text{Area of triangle} = \frac{1}{2} \times 6 \, \text{cm} \times 8 \, \text{cm} = \frac{1}{2} \times 48 \, \text{cm}^2 = 24 \, \text{cm}^2
\]

### Step 2: Area of each rectangular face

Next, we need to calculate the areas of the three rectangular faces. The dimensions given are:

Assuming the lengths of the rectangular faces correspond to the triangle sides and height of the prism (9 cm).

1. **Smallest rectangular face** (with side 6 cm):
\[
\text{Area} = \text{width} \times \text{height} = 6 \, \text{cm} \times 9 \, \text{cm} = 54 \, \text{cm}^2
\]

2. **Middle rectangular face** (with side 8 cm):
\[
\text{Area} = \text{width} \times \text{height} = 8 \, \text{cm} \times 9 \, \text{cm} = 72 \, \text{cm}^2
\]

3. **Largest rectangular face** (with side 10 cm):
\[
\text{Area} = \text{width} \times \text{height} = 10 \, \text{cm} \times 9 \, \text{cm} = 90 \, \text{cm}^2
\]

### Step 3: Calculate Total Surface Area

Now we need to sum the areas of the triangular faces and the rectangular faces.

- There are two triangular faces, so:
\[
\text{Total area of triangular faces} = 2 \times 24 \, \text{cm}^2 = 48 \, \text{cm}^2
\]

- Area of all three rectangular faces:
\[
\text{Total area of rectangular faces} = 54 \, \text{cm}^2 + 72 \, \text{cm}^2 + 90 \, \text{cm}^2 = 216 \, \text{cm}^2
\]

- Finally, add the triangular and rectangular areas together:
\[
\text{TOTAL Surface Area} = 48 \, \text{cm}^2 + 216 \, \text{cm}^2 = 264 \, \text{cm}^2
\]

### Final Answers:

- Area of each triangular face = **24 cm²**
- Area of the smallest rectangular face = **54 cm²**
- Area of the middle rectangular face = **72 cm²**
- Area of the largest rectangular face = **90 cm²**
- TOTAL Surface Area of the Triangular Prism = **264 cm²**