If the length of the diagonal of a cube is `8sqrt(3) cm`, then its surface area is

Options

A. `192 cm^(2)`
B. `512 cm^(2)`
C. `768 cm^(2)`
D. `384 cm^(2)`

Answer

To find the surface area of a cube when the length of its diagonal is given, we can follow these steps:

### Step 1: Understand the relationship between the diagonal and the side length 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: Set up the equation using the given diagonal length.
We know from the problem that the diagonal \( d \) is \( 8\sqrt{3} \) cm. Therefore, we can set up the equation:
\[ a\sqrt{3} = 8\sqrt{3} \]

### Step 3: Solve for the side length \( a \).
To find \( a \), we can divide both sides of the equation by \( \sqrt{3} \):
\[ a = 8 \text{ cm} \]

### Step 4: Calculate the surface area of the cube.
The formula for the surface area \( S \) of a cube is:
\[ S = 6a^2 \]
Substituting \( a = 8 \) cm into the formula:
\[ S = 6 \times (8)^2 \]
Calculating \( (8)^2 \):
\[ (8)^2 = 64 \]
Now substituting back into the surface area formula:
\[ S = 6 \times 64 = 384 \text{ cm}^2 \]

### Final Answer:
The surface area of the cube is \( 384 \text{ cm}^2 \).

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