To solve for the surface area of the triangular prism, we calculate the areas of all its faces:

Step 1: Identify the faces

The prism has:

1. Two triangular faces (front and back)
2. Three rectangular faces (sides, top, and bottom)

Step 2: Calculate the area of the triangular face

The triangle's base is $12 \, \text{cm}$ and its height is $16 \, \text{cm}$. The area of one triangle is: $\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}$ $\text{Area of triangle} = \frac{1}{2} \times 12 \times 16 = 96 \, \text{cm}^2$ Since there are two triangular faces: $\text{Total area of triangles} = 2 \times 96 = 192 \, \text{cm}^2$

Step 3: Calculate the area of the rectangular faces

1. Bottom face: The bottom face has a length of $20 \, \text{cm}$ and width $12 \, \text{cm}$: $\text{Area of bottom} = 20 \times 12 = 240 \, \text{cm}^2$
2. Slanted face: The slanted face has a length of $20 \, \text{cm}$ and width $16 \, \text{cm}$: $\text{Area of slanted face} = 20 \times 16 = 320 \, \text{cm}^2$
3. Vertical face: The vertical face has a length of $20 \, \text{cm}$ and width $9 \, \text{cm}$: $\text{Area of vertical face} = 20 \times 9 = 180 \, \text{cm}^2$

Step 4: Add up all the areas

$\text{Total Surface Area} = \text{Total area of triangles} + \text{Area of bottom} + \text{Area of slanted face} + \text{Area of vertical face}$ $\text{Total Surface Area} = 192 + 240 + 320 + 180 = 932 \, \text{cm}^2$

Final Answer:

The surface area of the triangular prism is: $\boxed{932 \, \text{cm}^2}$

Let me know if you'd like further details or have any questions!

Follow-up Questions:

1. How is the formula for the area of a triangle derived?
2. What are the key differences between calculating the surface area and the volume of a prism?
3. Could we solve this problem differently if the prism's dimensions were given in other units (e.g., meters)?
4. How do the lengths of the edges affect the prism's surface area?
5. What real-life objects resemble triangular prisms?

Tip:

Always double-check your dimensions to ensure they match when calculating areas for composite shapes!