Each face of a cube has perimeter equal to 32cm. Find its surface area and its volume.

Answer

To solve the problem step by step, we will first find the side length of the cube, then calculate its surface area and volume.

### Step 1: Find the side length of the cube
Given that the perimeter of each face of the cube is 32 cm, we know that each face of a cube is a square. The perimeter \( P \) of a square is given by the formula:

\[ P = 4 \times \text{side} \]

Setting the perimeter equal to 32 cm:

\[ 4 \times \text{side} = 32 \]

To find the side length, divide both sides by 4:

\[ \text{side} = \frac{32}{4} = 8 \text{ cm} \]

### Step 2: Calculate the surface area of the cube
The surface area \( A \) of a cube is given by the formula:

\[ A = 6 \times \text{side}^2 \]

Substituting the side length we found:

\[ A = 6 \times (8 \text{ cm})^2 \]

Calculating \( (8 \text{ cm})^2 \):

\[ (8 \text{ cm})^2 = 64 \text{ cm}^2 \]

Now substituting back into the surface area formula:

\[ A = 6 \times 64 \text{ cm}^2 = 384 \text{ cm}^2 \]

### Step 3: Calculate the volume of the cube
The volume \( V \) of a cube is given by the formula:

\[ V = \text{side}^3 \]

Substituting the side length:

\[ V = (8 \text{ cm})^3 \]

Calculating \( (8 \text{ cm})^3 \):

\[ (8 \text{ cm})^3 = 512 \text{ cm}^3 \]

### Final Results
- The surface area of the cube is \( 384 \text{ cm}^2 \).
- The volume of the cube is \( 512 \text{ cm}^3 \).

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