Understanding the Problem: Copper Wire Transformation

The question asks us to find the area of a circle that is formed by bending a copper wire. Initially, the same wire is bent into a square with a known area. The key concept here is that the length of the wire remains constant, regardless of whether it is shaped into a square or a circle. This means the perimeter of the square is equal to the circumference of the circle.

Step-by-Step Solution to find the Area of the Circle

1. Find the side length of the square

The area of the square is given as 121 cm². The formula for the area of a square is side × side, or \(s^2\), where \(s\) is the side length.

Given: Area of Square = 121 cm²

So, \(s^2 = 121\)

To find the side \(s\), we take the square root of the area:

\(s = \sqrt{121}\)

\(s = 11\) cm

The side length of the square is 11 cm.

2. Calculate the perimeter of the square

The perimeter of a square is given by the formula 4 × side, or \(4s\).

Perimeter of Square = \(4 \times s\)

Perimeter of Square = \(4 \times 11\) cm

Perimeter of Square = 44 cm

This perimeter is the total length of the copper wire.

3. Determine the circumference of the circle

Since the same wire is bent into a circle, the length of the wire is equal to the circumference of the circle.

Circumference of Circle = Length of Wire

Circumference of Circle = Perimeter of Square

Circumference of Circle = 44 cm

4. Find the radius of the circle

The formula for the circumference of a circle is \(2\pi r\), where \(r\) is the radius and \(\pi\) is approximately 22/7.

Given: Circumference of Circle = 44 cm

Using the formula: \(2\pi r = 44\)

Substitute the value of \(\pi = \frac{22}{7}\):

\(2 \times \frac{22}{7} \times r = 44\)

\(\frac{44}{7} \times r = 44\)

To find \(r\), multiply both sides by \(\frac{7}{44}\):

\(r = 44 \times \frac{7}{44}\)

\(r = 7\) cm

The radius of the circle is 7 cm.

5. Calculate the area of the circle

The formula for the area of a circle is \(\pi r^2\).

Area of Circle = \(\pi r^2\)

Substitute the value of \(\pi = \frac{22}{7}\) and \(r = 7\) cm:

Area of Circle = \(\frac{22}{7} \times (7)^2\)

Area of Circle = \(\frac{22}{7} \times 49\)

We can cancel out a 7 from the numerator and denominator:

Area of Circle = \(22 \times \frac{49}{7}\)

Area of Circle = \(22 \times 7\)

Area of Circle = 154 cm²

The area of the circle is 154 cm².

Revision Table: Geometry Formulas

Reviewing the key formulas used in this problem:

• Area of a Square: \(s^2\)
• Perimeter of a Square: \(4s\)
• Circumference of a Circle: \(2\pi r\)
• Area of a Circle: \(\pi r^2\)

Additional Information: Conservation of Length

This problem highlights the principle of conservation of length when a wire (or any flexible material) is reshaped. The total length of the material doesn't change, which allows us to equate the perimeter of the initial shape (square) to the circumference of the final shape (circle) to find missing dimensions like the radius.