The length of the diagonal of a cube is `6sqrt3` cm . Its total surface area is

Options

A. `144 cm^(2)`
B. `216 cm^(2)`
C. `180 cm^(2)`
D. `108 cm^(2)`

Explanation

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

### Step 1: Understand the relationship between the diagonal and the side of the cube.
The diagonal \(d\) of a cube with side length \(a\) can be calculated using the formula:
\[ d = a\sqrt{3} \]

### Step 2: Substitute the given diagonal length into the formula.
We are given that the diagonal \(d = 6\sqrt{3}\) cm. Therefore, we can set up the equation:
\[ 6\sqrt{3} = a\sqrt{3} \]

### Step 3: Solve for the side length \(a\).
To find \(a\), we can divide both sides of the equation by \(\sqrt{3}\):
\[ a = 6 \text{ cm} \]

### Step 4: Calculate the total surface area of the cube.
The total surface area \(A\) of a cube is given by the formula:
\[ A = 6a^2 \]
Substituting \(a = 6\) cm into the formula gives:
\[ A = 6 \times (6)^2 \]
\[ A = 6 \times 36 \]
\[ A = 216 \text{ cm}^2 \]

### Final Answer:
The total surface area of the cube is \(216 \text{ cm}^2\).

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