Let's solve the problem.

The problem states:

• The diagonal of a cube has a length of $\sqrt{12}$.
• We need to find the surface area of the cube.

Step 1: Formula for the space diagonal of a cube

The diagonal of a cube can be calculated using the formula: $d = a\sqrt{3}$ where $a$ is the side length of the cube, and $d$ is the diagonal.

Step 2: Solve for the side length

Here, the given diagonal $d = \sqrt{12}$. Substituting into the formula: $\sqrt{12} = a\sqrt{3}$ Solve for $a$ by dividing both sides by $\sqrt{3}$: $a = \frac{\sqrt{12}}{\sqrt{3}}$ Simplify $\frac{\sqrt{12}}{\sqrt{3}}$: $a = \sqrt{\frac{12}{3}} = \sqrt{4} = 2$ Thus, the side length of the cube is $a = 2$ units.

Step 3: Surface area of the cube

The surface area $S$ of a cube is given by: $S = 6a^2$ Substitute $a = 2$: $S = 6(2^2) = 6 \times 4 = 24$

Final Answer:

The surface area of the cube is: $\boxed{24 \text{ square units}}$

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

Related Questions:

1. How do you derive the formula for the space diagonal of a cube?
2. What is the volume of the cube if the side length is 2 units?
3. If the diagonal were $\sqrt{27}$, what would the surface area of the cube be?
4. How would this problem change if it were a rectangular prism instead of a cube?
5. Can the diagonal of a cube ever be an integer? Why or why not?

Tip:

To simplify square roots, try factoring into perfect squares (e.g., $\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}$).