Let’s solve the problem of finding the total surface area of the triangular prism step by step.

Step 1: Understand the given dimensions

The prism has the following measurements:

• The triangular face has sides 13 cm, 20 cm, and 12 cm (height).
• The rectangular faces are based on the prism's length of 30 cm.
• The base of the triangle is 16 cm.

Step 2: Calculate the area of the triangular face

The area of a triangle is given by the formula: $\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}$ Here: $\text{Base} = 16 \, \text{cm}, \quad \text{Height} = 12 \, \text{cm}$ $\text{Area of triangle} = \frac{1}{2} \times 16 \times 12 = 96 \, \text{cm}^2$

Step 3: Calculate the area of the three rectangular faces

Each rectangular face corresponds to one side of the triangle extended along the prism’s length of 30 cm.

1. First rectangle (side 13 cm): $\text{Area} = 13 \times 30 = 390 \, \text{cm}^2$

2. Second rectangle (side 20 cm): $\text{Area} = 20 \times 30 = 600 \, \text{cm}^2$

3. Third rectangle (base 16 cm): $\text{Area} = 16 \times 30 = 480 \, \text{cm}^2$

Step 4: Total surface area

The total surface area is the sum of:

• Two triangular faces (both are identical).
• Three rectangular faces.

$\text{Total surface area} = 2 \times \text{Area of triangle} + \text{Area of rectangles}$ $\text{Total surface area} = 2 \times 96 + 390 + 600 + 480$ $\text{Total surface area} = 192 + 390 + 600 + 480 = 1662 \, \text{cm}^2$

Final Answer:

The total surface area of the triangular prism is: $\boxed{1662 \, \text{cm}^2}$

Would you like further clarification or details on any part of this process?
Here are 5 related questions to expand your understanding:

1. How do you find the surface area of other types of prisms, such as rectangular or hexagonal prisms?
2. What is the relationship between volume and surface area of a prism?
3. How do you verify if a triangular prism is a right prism?
4. How would the solution change if the prism's length was different?
5. What are some real-life applications of calculating surface area for prisms?

Tip: Always break down the problem into simpler parts (e.g., calculate triangle areas and rectangle areas separately) to avoid mistakes!