Let's solve this step by step.

Problem Breakdown:

We are tasked with finding the total surface area of a triangular prism. A triangular prism has:

1. Two triangular bases.
2. Three rectangular lateral faces.

To find the total surface area: $\text{Total Surface Area} = \text{Area of the 2 Triangular Bases} + \text{Area of the 3 Rectangular Faces}.$

Step 1: Find the Area of the Triangular Base

The triangular base is a right triangle with dimensions $6 \, \text{cm}$ (base) and $8 \, \text{cm}$ (height). The area of a triangle is given by: $\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height}$ $\text{Area of one triangle} = \frac{1}{2} \times 6 \times 8 = 24 \, \text{cm}^2$ Since there are two triangular bases: $\text{Area of two triangles} = 2 \times 24 = 48 \, \text{cm}^2$

Step 2: Find the Area of the Rectangular Faces

The three rectangular faces are:

1. A rectangle with dimensions $10 \, \text{cm} \times 6 \, \text{cm}$: $\text{Area} = 10 \times 6 = 60 \, \text{cm}^2$
2. A rectangle with dimensions $10 \, \text{cm} \times 8 \, \text{cm}$: $\text{Area} = 10 \times 8 = 80 \, \text{cm}^2$
3. A rectangle with dimensions $10 \, \text{cm} \times 10 \, \text{cm}$: $\text{Area} = 10 \times 10 = 100 \, \text{cm}^2$

Step 3: Add All Areas Together

Now, sum up all the areas: $\text{Total Surface Area} = \text{Area of two triangles} + \text{Area of all rectangles}$ $\text{Total Surface Area} = 48 + 60 + 80 + 100 = 288 \, \text{cm}^2$

Final Answer:

$\boxed{288 \, \text{cm}^2}$

Would you like a detailed explanation of any step, or do you have additional questions?

Related Questions:

1. What is the formula for the volume of a triangular prism, and how does it compare to the surface area?
2. How do you calculate the diagonal length of a rectangular face in a prism?
3. Can you calculate the perimeter of the triangular base for this prism?
4. What would happen to the surface area if the dimensions of the prism were doubled?
5. How does the surface area of a cylinder differ from that of a prism with the same base?

Tip:

When solving surface area problems, always break down the shape into simpler components (triangles, rectangles, etc.) for easier calculations.