The diagonal of a cube is `sqrt12``"cm"^3`. Find its volume and surface area.

Answer

To find the volume and surface area of a cube given its diagonal, we can follow these steps:

### Step 1: Understand the relationship between the diagonal and the side length of a cube.
The formula for the diagonal \(d\) of a cube in terms of its side length \(s\) is given by:
\[ d = s \sqrt{3} \]

### Step 2: Set up the equation using the given diagonal.
We are given that the diagonal of the cube is \(\sqrt{12}\) cm. Therefore, we can set up the equation:
\[ \sqrt{12} = s \sqrt{3} \]

### Step 3: Solve for the side length \(s\).
To isolate \(s\), we can rearrange the equation:
\[ s = \frac{\sqrt{12}}{\sqrt{3}} \]
Now, simplify \(\sqrt{12}\):
\[ \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \]
Substituting this back into the equation gives:
\[ s = \frac{2\sqrt{3}}{\sqrt{3}} = 2 \text{ cm} \]

### Step 4: Calculate the volume of the cube.
The volume \(V\) of a cube is given by:
\[ V = s^3 \]
Substituting \(s = 2\) cm:
\[ V = 2^3 = 8 \text{ cm}^3 \]

### Step 5: Calculate the surface area of the cube.
The surface area \(A\) of a cube is given by:
\[ A = 6s^2 \]
Substituting \(s = 2\) cm:
\[ A = 6 \times (2^2) = 6 \times 4 = 24 \text{ cm}^2 \]

### Final Answers:
- Volume of the cube: \(8 \text{ cm}^3\)
- Surface area of the cube: \(24 \text{ cm}^2\)

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