Determining the Radius of a Circle from its Bending into a Square

Introduction: This article explores how the area of a square and the length of a copper wire can be used to determine the radius of a circle formed by bending the wire. The process involves understanding key mathematical concepts related to perimeter, area, and circumference.

Understanding the Problem

A copper wire is used to form a square with an area of 484 square centimeters (). The objective is to determine the radius of the circle that could be formed by the same wire if it is bent into that shape. This problem integrates concepts from both geometry and algebra.

Step-by-Step Solution

Step 1: Calculate the Length of the Wire

When the wire is bent into a square, the area A of the square is given as 484 cm2. The area of a square is given by the formula A s2, where s is the side length of the square.

s2 484

Solving for s, we get:

s sqrt{484} 22 , text{cm}

The perimeter of the square, which is the length of the wire, is given by:

P 4s 4 times 22 88 , text{cm}

The wire is 88 cm long.

Step 2: Calculate the Radius of the Circle

When the wire is bent into a circle, the circumference of the circle is equal to the length of the wire, which is 88 cm. The formula for the circumference of a circle is C 2pi r, where r is the radius of the circle.

2pi r 88

Solving for r using pi approx 3.14 , we get:

r frac{88}{2 times 3.14} frac{44}{3.14} approx 14.0 , text{cm}

Conclusion

Thus, the radius of the circle formed by bending the copper wire is approximately 14.0 cm. This calculation demonstrates how fundamental mathematical principles can be applied to solve real-world problems involving shapes and measurements.

Key Concepts and Formulas

Practical Applications

This problem-solving technique can be applied in various fields, such as engineering, architecture, and design. Understanding the relationships between the shapes and their measurements is crucial for solving complex problems and designing efficient structures and objects.

Additional Problem

For further practice, we can consider another scenario where the area of a square is 625 cm2. Using the same process, calculate the radius of the circle that could be formed with the same length of wire.