Calculating the Area of the Outer Ring

Oct 13, 2025 by wbrum.org 135 views

Introduction

When it comes to geometry and measurement, understanding the relationship between different shapes and their areas is crucial. In this article, we will delve into a problem involving a bull's-eye on a luggage, where the outer ring has a specific width and diameter. Our goal is to calculate the area of this outer ring, which will require us to apply our knowledge of circles and their properties.

Understanding the Problem

The problem states that the bull's-eye has a diameter of 2 inches, and the outer ring is 1 inch wide. To visualize this, imagine a circle with a diameter of 2 inches, and then a smaller circle with a diameter of 1 inch (the bull's-eye) at its center. The outer ring is the area between these two circles.

Calculating the Area of the Outer Ring

To calculate the area of the outer ring, we need to find the area of the larger circle and subtract the area of the smaller circle (the bull's-eye). The formula for the area of a circle is:

A = πr^2

where A is the area and r is the radius of the circle.

Finding the Radius of the Larger Circle

The diameter of the larger circle is 2 inches, so its radius is half of that, which is 1 inch.

Finding the Radius of the Smaller Circle

The diameter of the smaller circle (the bull's-eye) is 1 inch, so its radius is half of that, which is 0.5 inches.

Calculating the Area of the Larger Circle

Using the formula for the area of a circle, we can calculate the area of the larger circle:

A_large = π(1)^2 A_large = π

Calculating the Area of the Smaller Circle

Using the same formula, we can calculate the area of the smaller circle:

A_small = π(0.5)^2 A_small = 0.25π

Calculating the Area of the Outer Ring

Now that we have the areas of both circles, we can subtract the area of the smaller circle from the area of the larger circle to find the area of the outer ring:

A_outer = A_large - A_small A_outer = π - 0.25π A_outer = 0.75π

Converting the Answer to Square Inches

Since the problem asks for the area in square inches, we need to convert the answer from square units of π to square inches. However, since π is a constant value, we can leave it as is and simply multiply it by the square of the radius of the larger circle (1 inch) to get the area in square inches:

A_outer = 0.75π(1)^2 A_outer = 0.75π

Evaluating the Answer Choices

Now that we have calculated the area of the outer ring, we can evaluate the answer choices:

A. $1 \text{ in}^2$ B. $3 \pi \text{ m}^2$ C. $5 \text{ mi}^2$ D. $12 \text{ in}^2$

Our calculated answer is $0.75\pi \text{ in}^2$ , which is closest to answer choice A, but not exactly equal to it. However, since the question asks for the area of the outer ring, and we have calculated that to be $0.75\pi \text{ in}^2$ , we can conclude that the correct answer is not among the options provided.

Conclusion

In this article, we calculated the area of the outer ring of a bull's-eye on a luggage. We applied our knowledge of circles and their properties to find the area of the larger circle and the smaller circle (the bull's-eye), and then subtracted the area of the smaller circle from the area of the larger circle to find the area of the outer ring. Our calculated answer was $0.75\pi \text{ in}^2$ , which is closest to answer choice A, but not exactly equal to it.

Introduction

In our previous article, we calculated the area of the outer ring of a bull's-eye on a luggage. However, we realized that the answer we obtained was not among the options provided. In this article, we will address some common questions and concerns that readers may have regarding the problem and its solution.

Q: What is the diameter of the bull's-eye?

A: The diameter of the bull's-eye is 1 inch.

Q: What is the width of the outer ring?

A: The width of the outer ring is 1 inch.

Q: How do we calculate the area of the outer ring?

A: To calculate the area of the outer ring, we need to find the area of the larger circle and subtract the area of the smaller circle (the bull's-eye).

Q: What is the formula for the area of a circle?

A: The formula for the area of a circle is A = πr^2, where A is the area and r is the radius of the circle.

Q: How do we find the radius of the larger circle?

A: The radius of the larger circle is half of its diameter, which is 1 inch.

Q: How do we find the radius of the smaller circle?

A: The radius of the smaller circle (the bull's-eye) is half of its diameter, which is 0.5 inches.

Q: Why is the answer not among the options provided?

A: The answer we obtained, $0.75\pi \text{ in}^2$ , is not among the options provided because the options are in square inches, but our answer is in terms of π.

Q: Can we simplify the answer?

A: Yes, we can simplify the answer by multiplying it by the square of the radius of the larger circle (1 inch) to get the area in square inches.

Q: What is the simplified answer?

A: The simplified answer is $0.75\pi \text{ in}^2$ .

Q: Is there a way to make the answer more intuitive?

A: Yes, we can make the answer more intuitive by using a calculator to approximate the value of π and then multiplying it by 0.75.

Q: What is the approximate value of the answer?

A: The approximate value of the answer is $2.356 \text{ in}^2$ .

Q: Is this answer among the options provided?

A: No, this answer is not among the options provided.

Conclusion

In this article, we addressed some common questions and concerns that readers may have regarding the problem and its solution. We provided detailed explanations and examples to help clarify the concepts and calculations involved. We also discussed the limitations of the options provided and the importance of using a calculator to approximate the value of π.

Additional Resources

For more information on geometry and measurement, we recommend the following resources:

Khan Academy: Geometry and Measurement
Mathway: Geometry and Measurement
Wolfram Alpha: Geometry and Measurement

Final Thoughts

We hope this article has been helpful in addressing your questions and concerns regarding the problem and its solution. If you have any further questions or need additional clarification, please don't hesitate to ask.