Calculating the Area of the Label

Oct 13, 2025 by windailys.com 59 views

Introduction

In this article, we will explore the problem of calculating the area of a label on a cylindrical container. The container has a base diameter of 20 cm and a height of 27 cm. The label is placed 3 cm from the top of the container. We will use mathematical formulas to calculate the area of the label.

Understanding the Problem

The problem involves finding the area of a label on a cylindrical container. The container has a circular base with a diameter of 20 cm, which means the radius is 10 cm. The height of the container is 27 cm. The label is placed 3 cm from the top of the container, which means it covers a portion of the container's surface.

Calculating the Area of the Label

To calculate the area of the label, we need to find the area of the circular surface of the container and then subtract the area of the top and bottom circular surfaces that are not covered by the label.

Step 1: Calculate the Area of the Circular Surface

The area of a circular surface is given by the formula:

A = πr^2

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

In this case, the radius of the circular surface is 10 cm. Plugging in the value, we get:

A = π(10)^2 A = 314.16 cm^2

Step 2: Calculate the Area of the Top and Bottom Circular Surfaces

The area of the top and bottom circular surfaces is given by the same formula:

A = πr^2

In this case, the radius of the top and bottom circular surfaces is 10 cm. Plugging in the value, we get:

A = π(10)^2 A = 314.16 cm^2

Step 3: Calculate the Area of the Label

The area of the label is the difference between the area of the circular surface and the area of the top and bottom circular surfaces:

A_label = A - 2(A_top_bottom) A_label = 314.16 - 2(314.16) A_label = 314.16 - 628.32 A_label = -314.16

However, this is not the correct answer. We need to consider the fact that the label is placed 3 cm from the top of the container. This means that the label covers a portion of the circular surface, and we need to subtract the area of the top circular surface that is not covered by the label.

Step 4: Calculate the Area of the Top Circular Surface Not Covered by the Label

The area of the top circular surface not covered by the label is given by the formula:

A_top_not_covered = πr^2

In this case, the radius of the top circular surface not covered by the label is 10 cm. Plugging in the value, we get:

A_top_not_covered = π(10)^2 A_top_not_covered = 314.16 cm^2

Step 5: Calculate the Area of the Label

The area of the label is the difference between the area of the circular surface and the area of the top circular surface not covered by the label:

A_label = A - A_top_not_covered A_label = 314.16 - 314.16 A_label = 0

However, this is not the correct answer. We need to consider the fact that the label is placed 3 cm from the top of the container, and it covers a portion of the circular surface. We need to use the formula for the area of a circular segment to calculate the area of the label.

Calculating the Area of a Circular Segment

The area of a circular segment is given by the formula:

A_segment = (θ/360)πr^2 - (1/2)r^2sin(θ)

where A_segment is the area of the circular segment, θ is the central angle of the segment, and r is the radius of the circle.

In this case, the central angle of the segment is 360 - 2(90 - 3) = 78 degrees. Plugging in the values, we get:

A_segment = (78/360)π(10)^2 - (1/2)(10)^2sin(78) A_segment = 0.2169π(100) - 50sin(78) A_segment = 68.49 - 50(0.978) A_segment = 68.49 - 48.9 A_segment = 19.59

Conclusion

In this article, we calculated the area of a label on a cylindrical container. We used mathematical formulas to calculate the area of the circular surface, the area of the top and bottom circular surfaces, and the area of the label. We also used the formula for the area of a circular segment to calculate the area of the label. The final answer is 19.59 cm^2.

References

"Mathematics for Engineers and Scientists" by Donald R. Hill
"Calculus" by Michael Spivak
"Geometry" by I.M. Gelfand

Note

Introduction

In our previous article, we explored the problem of calculating the area of a label on a cylindrical container. We used mathematical formulas to calculate the area of the circular surface, the area of the top and bottom circular surfaces, and the area of the label. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the formula for calculating the area of a label on a cylindrical container?

A: The formula for calculating the area of a label on a cylindrical container is:

A_label = (θ/360)πr^2 - (1/2)r^2sin(θ)

where A_label is the area of the label, θ is the central angle of the segment, and r is the radius of the circle.

Q: What is the central angle of the segment?

A: The central angle of the segment is 360 - 2(90 - 3) = 78 degrees.

Q: How do I calculate the area of the circular surface?

A: The area of the circular surface is given by the formula:

A = πr^2

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

Q: How do I calculate the area of the top and bottom circular surfaces?

A: The area of the top and bottom circular surfaces is given by the same formula:

A = πr^2

Q: What is the radius of the top and bottom circular surfaces?

A: The radius of the top and bottom circular surfaces is 10 cm.

Q: How do I calculate the area of the label?

A: The area of the label is the difference between the area of the circular surface and the area of the top and bottom circular surfaces:

A_label = A - 2(A_top_bottom)

Q: What is the formula for the area of a circular segment?

A: The formula for the area of a circular segment is:

A_segment = (θ/360)πr^2 - (1/2)r^2sin(θ)

Q: How do I calculate the area of the label using the formula for the area of a circular segment?

A: To calculate the area of the label using the formula for the area of a circular segment, we need to plug in the values:

A_segment = (78/360)π(10)^2 - (1/2)(10)^2sin(78) A_segment = 0.2169π(100) - 50sin(78) A_segment = 68.49 - 50(0.978) A_segment = 68.49 - 48.9 A_segment = 19.59

Q: What is the final answer?

A: The final answer is 19.59 cm^2.

Q: What are some common mistakes to avoid when calculating the area of a label on a cylindrical container?

A: Some common mistakes to avoid when calculating the area of a label on a cylindrical container include:

Not considering the fact that the label is placed 3 cm from the top of the container
Not using the formula for the area of a circular segment to calculate the area of the label
Not plugging in the correct values into the formula

Conclusion

In this article, we answered some frequently asked questions related to calculating the area of a label on a cylindrical container. We provided formulas and examples to help you understand the concept. Remember to avoid common mistakes and use the correct formulas to calculate the area of the label.