To find the surface area of the triangular prism, we need to consider both the triangular faces and the rectangular faces of the prism.

Given dimensions:

• Triangle base: 6 units
• Triangle height: 4 units (perpendicular height of the triangle)
• Triangle side lengths: 5, 5, and 6 units
• Prism length (distance between triangular faces): 7 units

Step-by-Step Solution:

1. Calculate the Area of the Triangular Faces: The area of a triangle is given by: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ Substituting the values: $\text{Area of each triangle} = \frac{1}{2} \times 6 \times 4 = 12 \, \text{square units}$ Since there are two triangular faces, the total area of the triangular faces is: $2 \times 12 = 24 \, \text{square units}$

2. Calculate the Area of the Rectangular Faces: There are three rectangular faces, each with different dimensions:

   • Rectangle 1: $5 \times 7 = 35$
   • Rectangle 2: $5 \times 7 = 35$
   • Rectangle 3: $6 \times 7 = 42$

   Adding these areas together: $35 + 35 + 42 = 112 \, \text{square units}$

3. Add All Areas Together: Now, combine the areas of the triangular and rectangular faces: $24 + 112 = 136 \, \text{square units}$

Final Answer:

The surface area of the triangular prism is 136 square units.

Would you like more details or have any questions?

Related Questions:

1. How would the surface area change if the length of the prism increased?
2. What is the volume of this triangular prism?
3. How do we find the surface area of other prisms, such as rectangular or hexagonal prisms?
4. If only one side length of the triangle was given, how could we find the missing dimensions?
5. Can we use the Pythagorean theorem in any way here?

Tip:

When calculating surface area, always separate the different types of faces (triangular, rectangular, etc.) for clarity.