Answer :
To decide whether the wire bent into a square or into a circle has a larger area, let's go through the solution step-by-step.
1. Start with the wire's length:
The wire is 176 cm long.
2. Bend the wire into a square:
- Since a square has four equal sides, we divide the total length of the wire by 4 to find the length of one side:
[tex]\[ \text{Side of square} = \frac{176 \text{ cm}}{4} = 44 \text{ cm} \][/tex]
- The area of a square is given by the formula:
[tex]\[ \text{Area of square} = \text{side}^2 = 44^2 = 1936 \text{ square cm} \][/tex]
3. Bend the same wire into a circle:
- The circumference of the circle uses the entire length of the wire. The formula for the circumference of a circle is:
[tex]\[ \text{Circumference} = 2\pi \times \text{radius} \][/tex]
- Solving for the radius using the circumference:
[tex]\[ 176 = 2\pi \times \text{radius} \][/tex]
[tex]\[ \text{Radius} = \frac{176}{2\pi} \approx 28.011 \text{ cm} \][/tex]
- The area of a circle is calculated as:
[tex]\[ \text{Area of circle} = \pi \times \text{radius}^2 \approx \pi \times (28.011)^2 \approx 2464.992 \text{ square cm} \][/tex]
4. Comparison of areas:
- Area of the square = 1936 square cm
- Area of the circle = 2464.992 square cm
Since 2464.992 square cm (area of the circle) is greater than 1936 square cm (area of the square), the circle has more area.
In conclusion, when the wire is bent into a circle, it encloses a larger area than when it is bent into a square. Therefore, the circle will have more area.
1. Start with the wire's length:
The wire is 176 cm long.
2. Bend the wire into a square:
- Since a square has four equal sides, we divide the total length of the wire by 4 to find the length of one side:
[tex]\[ \text{Side of square} = \frac{176 \text{ cm}}{4} = 44 \text{ cm} \][/tex]
- The area of a square is given by the formula:
[tex]\[ \text{Area of square} = \text{side}^2 = 44^2 = 1936 \text{ square cm} \][/tex]
3. Bend the same wire into a circle:
- The circumference of the circle uses the entire length of the wire. The formula for the circumference of a circle is:
[tex]\[ \text{Circumference} = 2\pi \times \text{radius} \][/tex]
- Solving for the radius using the circumference:
[tex]\[ 176 = 2\pi \times \text{radius} \][/tex]
[tex]\[ \text{Radius} = \frac{176}{2\pi} \approx 28.011 \text{ cm} \][/tex]
- The area of a circle is calculated as:
[tex]\[ \text{Area of circle} = \pi \times \text{radius}^2 \approx \pi \times (28.011)^2 \approx 2464.992 \text{ square cm} \][/tex]
4. Comparison of areas:
- Area of the square = 1936 square cm
- Area of the circle = 2464.992 square cm
Since 2464.992 square cm (area of the circle) is greater than 1936 square cm (area of the square), the circle has more area.
In conclusion, when the wire is bent into a circle, it encloses a larger area than when it is bent into a square. Therefore, the circle will have more area.
