The lateral surface area of a cube us `256m^(2)`. The volume of the cube is.

Options

A. `512m^(3)`
B. `64m^(3)`
C. ` 216 m^(3)`
D. `256 m^(3)`

Explanation

To find the volume of the cube given its lateral surface area, we can follow these steps:

### Step 1: Understand the formula for the lateral surface area of a cube.
The lateral surface area (LSA) of a cube can be calculated using the formula:
\[ \text{LSA} = 4 \times \text{side}^2 \]

### Step 2: Set up the equation using the given lateral surface area.
We know from the problem that the lateral surface area is \(256 \, m^2\). Therefore, we can set up the equation:
\[ 4 \times \text{side}^2 = 256 \]

### Step 3: Solve for side squared.
To isolate \(\text{side}^2\), divide both sides of the equation by 4:
\[ \text{side}^2 = \frac{256}{4} \]
Calculating the right side gives:
\[ \text{side}^2 = 64 \]

### Step 4: Find the length of the side.
To find the length of the side, take the square root of both sides:
\[ \text{side} = \sqrt{64} \]
Calculating the square root gives:
\[ \text{side} = 8 \, m \]

### Step 5: Calculate the volume of the cube.
The volume \(V\) of a cube can be calculated using the formula:
\[ V = \text{side}^3 \]
Substituting the value of the side:
\[ V = 8^3 \]
Calculating \(8^3\):
\[ V = 8 \times 8 \times 8 = 512 \, m^3 \]

### Final Answer:
The volume of the cube is \(512 \, m^3\).

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