When each side of a cube is increased by 20%, then find the increase in total surface area of a cube .

Answer

To find the increase in the total surface area of a cube when each side is increased by 20%, we can follow these steps:

### Step 1: Understand the problem
We need to determine how much the total surface area of a cube increases when each side is increased by 20%.

**Hint:** Remember that the total surface area of a cube is calculated based on the length of its sides.

### Step 2: Define the original side length
Let the original side length of the cube be \( s \).

**Hint:** You can choose any value for \( s \) (like 1 unit) to simplify calculations, but it will cancel out later.

### Step 3: Calculate the new side length
If each side is increased by 20%, the new side length \( s' \) can be calculated as:
\[ s' = s + 0.2s = 1.2s \]

**Hint:** Increasing by 20% means you multiply the original length by \( 1 + 0.2 \).

### Step 4: Calculate the original total surface area
The total surface area \( A \) of a cube is given by the formula:
\[ A = 6s^2 \]

**Hint:** Remember that a cube has 6 faces, and the area of each face is \( s^2 \).

### Step 5: Calculate the new total surface area
Using the new side length \( s' \), the new total surface area \( A' \) is:
\[ A' = 6(s')^2 = 6(1.2s)^2 = 6 \times 1.44s^2 = 8.64s^2 \]

**Hint:** Make sure to square the new side length when calculating the area.

### Step 6: Calculate the increase in surface area
The increase in surface area \( \Delta A \) can be calculated as:
\[ \Delta A = A' - A = 8.64s^2 - 6s^2 = 2.64s^2 \]

**Hint:** Subtract the original surface area from the new surface area to find the increase.

### Step 7: Calculate the percentage increase
To find the percentage increase in surface area, use the formula:
\[ \text{Percentage Increase} = \left( \frac{\Delta A}{A} \right) \times 100 = \left( \frac{2.64s^2}{6s^2} \right) \times 100 \]
\[ = \left( \frac{2.64}{6} \right) \times 100 = 44\% \]

**Hint:** When calculating the percentage, the \( s^2 \) terms will cancel out.

### Final Answer
The increase in the total surface area of the cube is **44%**.