To solve this problem, we need to calculate both the surface area $S$ and the volume $V$ of the triangular prism, given that all edges of the prism measure $2\sqrt{6}$ units.

Step 1: Determine the Properties of the Prism

Since all edges are equal, the triangular base is an equilateral triangle with side lengths $2\sqrt{6}$. This means:

• The side length of the triangular base is $2\sqrt{6}$.
• The height of the prism (parallel edges) is also $2\sqrt{6}$.

Step 2: Surface Area of the Prism

The surface area $S$ of a prism is the sum of the areas of its two triangular bases and its three rectangular lateral faces.

1. Area of each triangular base:

   • For an equilateral triangle with side length $a$, the area $A$ is given by: $A = \frac{\sqrt{3}}{4} a^2$
   • Substituting $a = 2\sqrt{6}$: $A = \frac{\sqrt{3}}{4} (2\sqrt{6})^2 = \frac{\sqrt{3}}{4} \times 24 = 6\sqrt{3}$
   • Since there are two triangular bases, the total area for the bases is: $2 \times 6\sqrt{3} = 12\sqrt{3}$

2. Area of the three rectangular lateral faces:

   • Each rectangle has one side as the height of the prism and the other as the side of the triangular base, both measuring $2\sqrt{6}$.
   • The area of one rectangle is: $2\sqrt{6} \times 2\sqrt{6} = 24$
   • Since there are three such rectangles, the total area for the lateral faces is: $3 \times 24 = 72$

3. Total Surface Area: $S = 12\sqrt{3} + 72$

Step 3: Volume of the Prism

The volume $V$ of a prism is given by the area of the base times the height of the prism.

1. Volume Calculation:
   • The area of the triangular base is $6\sqrt{3}$ (as calculated above).
   • The height of the prism is $2\sqrt{6}$.
   • Therefore, the volume $V$ is: $V = 6\sqrt{3} \times 2\sqrt{6} = 12\sqrt{18} = 36\sqrt{2}$

Final Answers

• Surface Area: $S = 12\sqrt{3} + 72$
• Volume: $V = 36\sqrt{2}$

Would you like further details on any step or explanation?

Here are five related questions to deepen your understanding:

1. How would the surface area change if the height of the prism were doubled?
2. How do you find the area of an equilateral triangle given its side length?
3. What is the formula for the volume of any prism, and why does it work?
4. How can we generalize this solution to prisms with non-equilateral triangular bases?
5. What if the side length had been given in a different radical form? How would that affect simplification?

Tip: Remember that the area of an equilateral triangle depends only on its side length, so using the standard formula can save time in future calculations.