The diagonal of a cube is `16sqrt3` cm. Find its surface area and volume.

Answer

To solve the problem, we need to find the surface area and volume of a cube given that its diagonal is \(16\sqrt{3}\) cm.

### Step-by-Step Solution:

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:
\[ d = a\sqrt{3} \]

2. **Set up the equation using the given diagonal:**
We know from the problem that the diagonal \(d\) is \(16\sqrt{3}\) cm. Therefore, we can set up the equation:
\[ 16\sqrt{3} = a\sqrt{3} \]

3. **Solve for the side length \(a\):**
To find \(a\), we can divide both sides of the equation by \(\sqrt{3}\):
\[ a = 16 \]

4. **Calculate the surface area of the cube:**
The formula for the surface area \(SA\) of a cube is:
\[ SA = 6a^2 \]
Substituting \(a = 16\):
\[ SA = 6 \times (16)^2 \]
\[ SA = 6 \times 256 = 1536 \text{ cm}^2 \]

5. **Calculate the volume of the cube:**
The formula for the volume \(V\) of a cube is:
\[ V = a^3 \]
Substituting \(a = 16\):
\[ V = (16)^3 \]
\[ V = 4096 \text{ cm}^3 \]

### Final Answers:
- Surface Area = \(1536 \text{ cm}^2\)
- Volume = \(4096 \text{ cm}^3\)