To solve for the surface area of the triangular prism shown in the image, we can use the following steps:

1. Understand the prism structure:

   • It consists of two triangular bases and three rectangular faces.

2. Calculate the area of the triangular base:

   • The base is a right triangle with dimensions $3 \, \text{cm}$, $4 \, \text{cm}$, and hypotenuse $5 \, \text{cm}$.
   • Area of one triangular base:
     $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \times 4 = 6 \, \text{cm}^2.$

3. Calculate the lateral surface areas:

   • There are three rectangular faces:
     1. Rectangle 1: $3 \, \text{cm} \times 2 \, \text{cm}$, so area = $3 \times 2 = 6 \, \text{cm}^2$.
     2. Rectangle 2: $4 \, \text{cm} \times 2 \, \text{cm}$, so area = $4 \times 2 = 8 \, \text{cm}^2$.
     3. Rectangle 3: $5 \, \text{cm} \times 2 \, \text{cm}$, so area = $5 \times 2 = 10 \, \text{cm}^2$.

4. Sum the areas:

   • Total surface area = $2 \times \text{area of triangular base} + \text{areas of rectangles}$.
   • Total surface area = $2 \times 6 + 6 + 8 + 10 = 12 + 24 = 36 \, \text{cm}^2$.

Thus, the surface area of the triangular prism is 36 cm².

Would you like a detailed breakdown of any specific part of the calculation? Here are related questions for further understanding:

1. How is the formula for the area of a triangle derived?
2. Why does the prism have two triangular bases and three rectangular faces?
3. How can you find the perimeter of the prism for different purposes?
4. What is the volume of this triangular prism?
5. How does increasing the height (2 cm) affect the surface area?

Tip: Always verify dimensions for clarity and units consistency when working with 3D shapes.