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

Options

A. `1127cm^(3),765cm^(2)`
B. `1728cm^(3),864cm^(2)`
C. `1540cm^(3),820cm^(2)`
D. None of these

Answer

To solve the problem, we need to find the volume and surface area of a cube given its diagonal. Let's go through the steps systematically.

### Step 1: Understand 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:
\[ d = a \sqrt{3} \]

### Step 2: Substitute the given diagonal into the formula.
We are given that the diagonal \(d = 12\sqrt{3}\) cm. Substituting this into the formula:
\[ 12\sqrt{3} = a \sqrt{3} \]

### Step 3: Solve for the side length \(a\).
To find the side length \(a\), divide both sides by \(\sqrt{3}\):
\[ a = \frac{12\sqrt{3}}{\sqrt{3}} = 12 \text{ cm} \]

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

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

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