Cutting the Circle into 4 Equal Parts

Oct 13, 2025 by wbrum.org 116 views

Introduction

In this article, we will explore the concept of a circle being cut into 4 equal parts and rearranged to form another shape. We will analyze the perimeter of the original circle and the resulting shape to determine if there is a change in the perimeter. If so, we will calculate the amount of increase or decrease.

The Original Circle

A circle is a closed shape with no beginning or end, where every point on the circle is equidistant from a fixed central point called the center. The distance from the center to any point on the circle is called the radius. In this case, the radius of the circle is 16 cm.

Cutting the Circle into 4 Equal Parts

To cut the circle into 4 equal parts, we need to divide it into 4 equal arcs. Each arc will have a central angle of 90 degrees (or π/2 radians). This can be done by drawing 4 lines from the center of the circle to the circumference, each line passing through the center of one of the 4 equal parts.

Rearranging the Parts to Form a New Shape

Once the circle is cut into 4 equal parts, we can rearrange them to form a new shape. In this case, we will arrange the 4 parts to form a square. Each part will be a quarter of the original circle, and when arranged together, they will form a square with 4 equal sides.

Calculating the Perimeter of the Original Circle

The perimeter of a circle is given by the formula:

P = 2πr

where P is the perimeter and r is the radius of the circle.

Substituting the value of the radius (16 cm) into the formula, we get:

P = 2π(16) P = 32π P ≈ 100.53 cm

Calculating the Perimeter of the New Shape (Square)

The perimeter of a square is given by the formula:

P = 4s

where P is the perimeter and s is the length of one side of the square.

Since each part of the circle is a quarter of the original circle, the length of one side of the square will be equal to the radius of the circle, which is 16 cm.

Substituting the value of the side length into the formula, we get:

P = 4(16) P = 64 cm

Comparing the Perimeters

Now that we have calculated the perimeters of both shapes, we can compare them to determine if there is a change in the perimeter.

The original circle has a perimeter of approximately 100.53 cm, while the new shape (square) has a perimeter of 64 cm.

Conclusion

The perimeter of the original circle is approximately 100.53 cm, while the perimeter of the new shape (square) is 64 cm. This means that the perimeter of the new shape is 35.53 cm less than the original circle.

In other words, the perimeter of the new shape decreases by 35.53% compared to the original circle.

Key Takeaways

The perimeter of a circle is given by the formula P = 2πr.
The perimeter of a square is given by the formula P = 4s.
When a circle is cut into 4 equal parts and rearranged to form a square, the perimeter of the new shape decreases by approximately 35.53% compared to the original circle.

Final Thoughts

Q: What happens to the perimeter of a circle when it is cut into 4 equal parts and rearranged to form a square?

A: The perimeter of the new shape (square) decreases by approximately 35.53% compared to the original circle.

Q: Why does the perimeter of the square decrease compared to the original circle?

A: The perimeter of the square decreases because the square has a smaller perimeter due to the rearrangement of the circle's parts. When the circle is cut into 4 equal parts, each part is a quarter of the original circle. When these parts are rearranged to form a square, the resulting shape has a smaller perimeter.

Q: Can the perimeter of the square be greater than the original circle?

A: No, the perimeter of the square cannot be greater than the original circle. The perimeter of the square is always less than the original circle when the circle is cut into 4 equal parts and rearranged to form a square.

Q: What is the formula for calculating the perimeter of a circle?

A: The formula for calculating the perimeter of a circle is P = 2πr, where P is the perimeter and r is the radius of the circle.

Q: What is the formula for calculating the perimeter of a square?

A: The formula for calculating the perimeter of a square is P = 4s, where P is the perimeter and s is the length of one side of the square.

Q: Can the perimeter of the square be equal to the original circle?

A: No, the perimeter of the square cannot be equal to the original circle. The perimeter of the square is always less than the original circle when the circle is cut into 4 equal parts and rearranged to form a square.

Q: What is the significance of understanding the properties of shapes and how they change when transformed?

A: Understanding the properties of shapes and how they change when transformed is significant because it helps us to analyze and solve problems involving geometry and spatial reasoning. It also helps us to develop critical thinking skills and to visualize complex shapes and their transformations.

Q: Can the concept of cutting a circle into 4 equal parts and rearranging it to form a square be applied to other shapes?

A: Yes, the concept of cutting a circle into 4 equal parts and rearranging it to form a square can be applied to other shapes. However, the resulting shape and its perimeter will depend on the specific shape and its properties.

Q: What are some real-world applications of understanding the properties of shapes and how they change when transformed?

A: Some real-world applications of understanding the properties of shapes and how they change when transformed include:

Architecture: Understanding the properties of shapes and how they change when transformed is essential in architecture, where buildings and structures are designed and constructed.
Engineering: Understanding the properties of shapes and how they change when transformed is essential in engineering, where machines and mechanisms are designed and constructed.
Art and Design: Understanding the properties of shapes and how they change when transformed is essential in art and design, where shapes and forms are used to create visual compositions.

Conclusion

In conclusion, the perimeter of a circle decreases when it is cut into 4 equal parts and rearranged to form a square. Understanding the properties of shapes and how they change when transformed is significant in various fields, including architecture, engineering, and art and design.