To find the surface area of the triangular prism, we first need to determine its dimensions and use those to calculate the total surface area. Let’s solve step by step:

Step 1: Understanding the total edge length

A regular triangular prism has:

• 3 edges for the triangular base,
• 3 edges for the other triangular base, and
• 3 vertical edges connecting the two triangular bases.

Thus, there are a total of $3 + 3 + 3 = 9$ edges.

Given that the total edge length is $36$, the length of one edge of the triangular base can be calculated as follows: $3 \cdot \text{side of base} + 3 \cdot \text{height} = 36$ Substitute the given height ($h = 6$): $3s + 3(6) = 36$ $3s + 18 = 36$ $3s = 18 \quad \Rightarrow \quad s = 6$ So, the side length of the equilateral triangle is $s = 6$.

Step 2: Surface area of the prism

The surface area of a triangular prism is given by: $\text{Surface Area} = 2 \cdot \text{Base Area} + \text{Lateral Area}$

(a) Area of the triangular base

The formula for the area of an equilateral triangle is: $\text{Base Area} = \frac{\sqrt{3}}{4} s^2$ Substitute $s = 6$: $\text{Base Area} = \frac{\sqrt{3}}{4} (6^2) = \frac{\sqrt{3}}{4} \cdot 36 = 9\sqrt{3}$

(b) Lateral area

The lateral area is the sum of the areas of the three rectangular sides. Each rectangle has a base $s = 6$ and height $h = 6$: $\text{Area of one rectangle} = s \cdot h = 6 \cdot 6 = 36$ $\text{Lateral Area} = 3 \cdot 36 = 108$

(c) Total surface area

Now, substitute the values: $\text{Surface Area} = 2 \cdot \text{Base Area} + \text{Lateral Area}$ $\text{Surface Area} = 2 \cdot 9\sqrt{3} + 108$ $\text{Surface Area} = 18\sqrt{3} + 108$

Thus, the surface area of the triangular prism is: $\boxed{18\sqrt{3} + 108 \, \text{square units}}$

Do you want further details or have any questions?

Here are 5 related questions to explore:

1. How is the formula for the area of an equilateral triangle derived?
2. What changes if the prism is not regular but scalene?
3. Can you calculate the volume of this prism using the given dimensions?
4. How would the surface area change if the height doubled?
5. What is the significance of the lateral area in practical applications?

Tip: Always confirm the dimensions are consistent (e.g