A wire is in the shape of a square of side 10 cm. If the wire is rebent into a rectangle of length 12 cm, find its breadth. Which figure encloses more area and by how much?

Answer

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

### Step 1: Calculate the perimeter of the square.
The side of the square is given as 10 cm. The formula for the perimeter (P) of a square is:
\[ P = 4 \times \text{side} \]
Substituting the value:
\[ P = 4 \times 10 = 40 \text{ cm} \]

### Step 2: Set the perimeter of the rectangle equal to the perimeter of the square.
Since the wire is rebent into a rectangle, the perimeter of the rectangle will also be 40 cm. The formula for the perimeter (P) of a rectangle is:
\[ P = 2 \times (L + B) \]
Where L is the length and B is the breadth. We know the length (L) is 12 cm, so we can set up the equation:
\[ 2 \times (12 + B) = 40 \]

### Step 3: Solve for the breadth (B) of the rectangle.
First, divide both sides of the equation by 2:
\[ 12 + B = \frac{40}{2} \]
\[ 12 + B = 20 \]
Now, subtract 12 from both sides:
\[ B = 20 - 12 \]
\[ B = 8 \text{ cm} \]

### Step 4: Calculate the area of the square.
The formula for the area (A) of a square is:
\[ A = \text{side}^2 \]
Substituting the value:
\[ A = 10^2 = 100 \text{ cm}^2 \]

### Step 5: Calculate the area of the rectangle.
The formula for the area (A) of a rectangle is:
\[ A = L \times B \]
Substituting the values:
\[ A = 12 \times 8 = 96 \text{ cm}^2 \]

### Step 6: Compare the areas of the square and rectangle.
Now we compare the areas:
- Area of square = 100 cm²
- Area of rectangle = 96 cm²

To find out which figure encloses more area and by how much:
\[ \text{Difference} = 100 - 96 = 4 \text{ cm}^2 \]

### Conclusion:
The square encloses more area than the rectangle by 4 cm².

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