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To calculate the surface area of a triangular prism, we need to break it down into its individual surfaces. The prism consists of two triangular bases and three rectangular faces. Let's analyze the given dimensions:

- The triangular base has sides of 9 cm, 12 cm, and 16 cm.
- The height of the prism is 20 cm (distance between the two triangular bases).

### Step 1: Calculate the area of the triangular base
We will use **Heron's formula** to calculate the area of the triangle. The formula is:

\[
A_{\text{triangle}} = \sqrt{s(s-a)(s-b)(s-c)}
\]
Where \(a = 9 \, \text{cm}\), \(b = 12 \, \text{cm}\), and \(c = 16 \, \text{cm}\), and \(s\) is the semi-perimeter:
\[
s = \frac{a + b + c}{2} = \frac{9 + 12 + 16}{2} = 18.5 \, \text{cm}
\]
Now we can calculate the area of the triangle.

### Step 2: Calculate the area of the rectangular faces
The three rectangular faces have the following dimensions:
- One face: \(9 \, \text{cm} \times 20 \, \text{cm}\)
- Second face: \(12 \, \text{cm} \times 20 \, \text{cm}\)
- Third face: \(16 \, \text{cm} \times 20 \, \text{cm}\)

### Step 3: Calculate the total surface area
The total surface area of the prism is the sum of the areas of the two triangular bases and the three rectangular faces.

Let's calculate all these values.The total surface area of the triangular prism is approximately \( 846.88 \, \text{cm}^2 \).

Would you like further details on any of the calculations, or do you have any other questions?

Here are 5 related questions you might find helpful:
1. How does Heron's formula work for different types of triangles?
2. How do you calculate the volume of a triangular prism?
3. What other methods can be used to find the area of a triangle?
4. How is surface area important in real-world applications?
5. Can you use the Pythagorean theorem to verify if the triangle in the prism is a right triangle?

**Tip:** For complex shapes, break them down into simpler components (like triangles and rectangles) to make surface area calculations easier.

Work out the surface area of the triangular prism shown in the image.

Learn how to calculate the surface area of a triangular prism using Heron's formula for the triangle's base and standard area calculations for rectangles. This problem is suitable for students in Grades 8-10 and covers key geometry concepts and formulas.

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