Deepak took a wire of length 176 cm and bent it into the shape of a circle. Find the radius of that circle. Also, tind its area. IF the sam wire is bent into the shape of a rectangle of length 7 cm, what will be the breadth of the rectangle ? Which encloses more area, the circle or the rectangle ?

Answer

To solve the problem step by step, we will break it down into parts as follows:

### Step 1: Find the radius of the circle
1. **Understanding the problem**: The wire of length 176 cm is bent into a circle. The circumference of the circle is equal to the length of the wire.

2. **Formula for circumference**: The formula for the circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
where \( r \) is the radius.

3. **Setting up the equation**: Since the circumference is equal to the length of the wire:
\[ 2\pi r = 176 \]

4. **Solving for \( r \)**:
\[ r = \frac{176}{2\pi} \]
Using \( \pi \approx \frac{22}{7} \):
\[ r = \frac{176}{2 \times \frac{22}{7}} = \frac{176 \times 7}{44} = \frac{1232}{44} = 28 \text{ cm} \]

### Step 2: Find the area of the circle
1. **Formula for area**: The area \( A \) of a circle is given by:
\[ A = \pi r^2 \]

2. **Substituting the value of \( r \)**:
\[ A = \pi (28)^2 = \frac{22}{7} \times 784 \]

3. **Calculating the area**:
\[ A = 22 \times 112 = 2464 \text{ cm}^2 \]

### Step 3: Find the breadth of the rectangle
1. **Understanding the rectangle**: The same wire is now bent into the shape of a rectangle with a length of 7 cm. The perimeter of the rectangle is equal to the length of the wire.

2. **Formula for perimeter**: The perimeter \( P \) of a rectangle is given by:
\[ P = 2(l + b) \]
where \( l \) is the length and \( b \) is the breadth.

3. **Setting up the equation**:
\[ 2(7 + b) = 176 \]

4. **Solving for \( b \)**:
\[ 7 + b = \frac{176}{2} = 88 \]
\[ b = 88 - 7 = 81 \text{ cm} \]

### Step 4: Compare the areas of the circle and rectangle
1. **Area of the rectangle**:
\[ \text{Area} = l \times b = 7 \times 81 = 567 \text{ cm}^2 \]

2. **Comparison**:
- Area of the circle = 2464 cm²
- Area of the rectangle = 567 cm²

3. **Conclusion**: The circle encloses more area than the rectangle.

### Final Answers:
- Radius of the circle: 28 cm
- Area of the circle: 2464 cm²
- Breadth of the rectangle: 81 cm
- The circle encloses more area than the rectangle.