The diagonal of a cube is `6 sqrt3` cm. Find its volume.

Options

A. 612 `cm^3`
B. 216 `cm^3`
C. 226 `cm^3`
D. 136 `cm^3`

Answer

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

### 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 know the diagonal \(d\) is \(6\sqrt{3}\) cm. So 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 \(\sqrt{3}\):
\[ a = \frac{6\sqrt{3}}{\sqrt{3}} = 6 \text{ cm} \]

### Step 4: Calculate the volume of the cube.
The volume \(V\) of a cube is given by the formula:
\[ V = a^3 \]
Substituting \(a = 6\) cm into the formula:
\[ V = 6^3 = 216 \text{ cm}^3 \]

### Final Answer:
The volume of the cube is \(216 \text{ cm}^3\).

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