The diagonal of a cube is `12 sqrt(3)` cm. Its volume and surface area would be

Options

A. `112 cm^(2) , 765 cm ^(2)`
B. `1728 cm ^(2), 864 cm ^(2)`
C. `1540 cm ^(3) , 820 cm ^(2)`
D. none of these

Answer

To solve the problem, we need to find the volume and surface area of a cube given that the diagonal of the cube is \(12 \sqrt{3}\) cm.

### Step-by-Step Solution:

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

2. **Set up the equation using the given diagonal**:
We know that the diagonal \(d\) is \(12 \sqrt{3}\) cm. Therefore, we can set up the equation:
\[ 12 \sqrt{3} = a \sqrt{3} \]

3. **Solve for the side length \(a\)**:
To find \(a\), we can divide both sides of the equation by \(\sqrt{3}\):
\[ a = 12 \text{ cm} \]

4. **Calculate the volume of the cube**:
The volume \(V\) of a cube is given by the formula:
\[ V = a^3 \]
Substituting the value of \(a\):
\[ V = 12^3 = 1728 \text{ cm}^3 \]

5. **Calculate the surface area of the cube**:
The surface area \(SA\) of a cube is given by the formula:
\[ SA = 6a^2 \]
Substituting the value of \(a\):
\[ SA = 6 \times (12^2) = 6 \times 144 = 864 \text{ cm}^2 \]

### Final Answers:
- Volume of the cube: \(1728 \text{ cm}^3\)
- Surface area of the cube: \(864 \text{ cm}^2\)

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