Understanding the Geometry Problem: Copper Wire Transformation

This problem involves a copper wire that changes shape from a square to a circle. The key concept here is that the length of the wire remains constant regardless of its shape. The length of the wire when bent into a square is its perimeter, and when bent into a circle, it's its circumference.

Step-by-Step Solution for Area Calculation

1. Finding 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 multiplied by side, or side squared (\(s^2\)).

Let \(s\) be the side length of the square.

Area of square \(= s^2\)

\(121 \text{ cm}^2 = s^2\)

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

\(s = \sqrt{121 \text{ cm}^2}\)

\(s = 11 \text{ cm}\)

The side length of the square is 11 cm.

2. Calculating the Perimeter of the Square

The perimeter of a square is the total length of its boundary, which is 4 times the side length.

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

Perimeter of square \(= 4 \times 11 \text{ cm}\)

Perimeter of square \(= 44 \text{ cm}\)

This perimeter is the total length of the copper wire.

3. Relating Wire Length to Circle Circumference

When the same wire is bent into a circle, its length becomes the circumference of the circle.

Length of wire \(=\) Circumference of circle

Circumference of circle \(= 44 \text{ cm}\)

4. Finding the Radius of the Circle

The formula for the circumference of a circle is \(2\pi r\), where \(r\) is the radius of the circle. We can use the wire length (circumference) to find the radius.

Circumference \(= 2\pi r\)

\(44 \text{ cm} = 2 \times \pi \times r\)

Using the value of \(\pi \approx \frac{22}{7}\):

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

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

To find \(r\), we rearrange the equation:

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

\(r = 7 \text{ cm}\)

The radius of the circle is 7 cm.

5. Calculating the Area of the Circle

Now that we have the radius, we can calculate the area of the circle using the formula Area of circle \(= \pi r^2\).

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

Area of circle \(= \frac{22}{7} \times (7 \text{ cm})^2\)

Area of circle \(= \frac{22}{7} \times 49 \text{ cm}^2\)

Area of circle \(= 22 \times \frac{49}{7} \text{ cm}^2\)

Area of circle \(= 22 \times 7 \text{ cm}^2\)

Area of circle \(= 154 \text{ cm}^2\)

Final Answer

The area enclosed by the wire when bent in the form of a circle is 154 cm².

This matches one of the given options.

Revision Table: Key Concepts & Formulas

Additional Information: Shape Transformations & Area

This problem illustrates an important principle in geometry: changing the shape of a fixed-length perimeter affects the enclosed area. For a given perimeter, the shape that encloses the maximum area is a circle. A square encloses a smaller area than a circle of the same perimeter.

• When a wire (fixed length) is reshaped, its total length (perimeter or circumference) stays the same.
• The area enclosed by different shapes made from the same length of wire will generally be different.
• A circle is often referred to as the most efficient shape in terms of enclosing the maximum area for a given perimeter.