A solid cube of side 12 cm is out into 8 identical cubes. What will be the side of the new cube ? Also . Find the ratio between the surface area of the original cube and the total surface area of all the small cubes formed.

Answer

To solve the problem step-by-step, we will follow these instructions:

### Step 1: Find the side of the new cube
1. **Given**: The side of the original cube is 12 cm.
2. **Volume of the original cube**:
\[ V = \text{side}^3 = 12^3 = 1728 \text{ cm}^3 \]
3. **Volume of one small cube**: Since the original cube is cut into 8 identical cubes, the volume of one small cube is:
\[ V_{\text{small}} = \frac{V_{\text{original}}}{8} = \frac{1728}{8} = 216 \text{ cm}^3 \]
4. **Let \( x \) be the side of the small cube**. Then:
\[ x^3 = 216 \]
5. **Finding \( x \)**:
\[ x = \sqrt[3]{216} = 6 \text{ cm} \]

### Step 2: Find the surface area of the original cube
1. **Surface area of the original cube**:
\[ A = 6 \times \text{side}^2 = 6 \times 12^2 = 6 \times 144 = 864 \text{ cm}^2 \]

### Step 3: Find the total surface area of all small cubes
1. **Surface area of one small cube**:
\[ A_{\text{small}} = 6 \times \text{side}^2 = 6 \times 6^2 = 6 \times 36 = 216 \text{ cm}^2 \]
2. **Total surface area of all 8 small cubes**:
\[ B = 8 \times A_{\text{small}} = 8 \times 216 = 1728 \text{ cm}^2 \]

### Step 4: Find the ratio of the surface area of the original cube to the total surface area of the small cubes
1. **Ratio**:
\[ \text{Ratio} = \frac{A}{B} = \frac{864}{1728} = \frac{1}{2} \]

### Final Answers:
- The side of the new cube is **6 cm**.
- The ratio of the surface area of the original cube to the total surface area of all small cubes is **1:2**.

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