The Lateral Surface Area of a Hollow Cylinder

Oct 13, 2025 by miodeya.com 58 views

Introduction

In this article, we will explore the relationship between the lateral surface area of a hollow cylinder and the perimeter of a rectangular sheet formed when the cylinder is cut along its height. We will use mathematical concepts to derive the solution and provide a step-by-step guide to calculate the perimeter of the rectangular sheet.

The Lateral Surface Area of a Hollow Cylinder

The lateral surface area of a hollow cylinder is given by the formula:

LSA = 2πrh

where LSA is the lateral surface area, π is a mathematical constant approximately equal to 3.14, r is the radius of the cylinder, and h is the height of the cylinder.

Given that the lateral surface area of the hollow cylinder is 4224 cm², we can set up the equation:

2πrh = 4224

Converting the Equation to a Solvable Form

To solve for the radius (r) and height (h) of the cylinder, we need to isolate these variables in the equation. We can start by dividing both sides of the equation by 2π:

rh = 4224 / (2π)

rh = 4224 / (2 × 3.14)

rh = 4224 / 6.28

rh = 672

Finding the Relationship Between the Radius and Height

Now that we have the product of the radius and height, we can use this information to find the relationship between the two variables. Since we know that the product of the radius and height is 672, we can write:

r × h = 672

The Relationship Between the Lateral Surface Area and the Rectangular Sheet

When the cylinder is cut along its height, it forms a rectangular sheet with a width of 33 cm. The length of the rectangular sheet is equal to the height of the cylinder (h). The perimeter of the rectangular sheet is given by the formula:

P = 2(l + w)

where P is the perimeter, l is the length, and w is the width.

Finding the Perimeter of the Rectangular Sheet

Since the width of the rectangular sheet is 33 cm, we can substitute this value into the formula for the perimeter:

P = 2(l + 33)

We also know that the length of the rectangular sheet is equal to the height of the cylinder (h). To find the height of the cylinder, we can use the relationship between the radius and height:

r × h = 672

Solving for the Height of the Cylinder

We can rearrange the equation to solve for the height (h):

h = 672 / r

Finding the Radius of the Cylinder

We can use the lateral surface area equation to find the radius of the cylinder:

2πrh = 4224

Substituting the Value of h

We can substitute the expression for h into the lateral surface area equation:

2πr(672 / r) = 4224

Simplifying the Equation

We can simplify the equation by canceling out the r term:

2π(672) = 4224

Solving for the Radius

We can solve for the radius (r) by dividing both sides of the equation by 2π:

r = 4224 / (2π × 672)

r = 4224 / (2 × 3.14 × 672)

r = 4224 / 4223.68

r = 1

Finding the Height of the Cylinder

Now that we have the radius of the cylinder, we can find the height (h) by substituting the value of r into the equation:

h = 672 / r

h = 672 / 1

h = 672

Finding the Perimeter of the Rectangular Sheet

Now that we have the length and width of the rectangular sheet, we can find the perimeter by substituting these values into the formula:

P = 2(l + w)

P = 2(672 + 33)

P = 2(705)

P = 1410

The final answer is: 1410

Introduction

In our previous article, we explored the relationship between the lateral surface area of a hollow cylinder and the perimeter of a rectangular sheet formed when the cylinder is cut along its height. We provided a step-by-step guide to calculate the perimeter of the rectangular sheet. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the lateral surface area of a hollow cylinder?

A: The lateral surface area of a hollow cylinder is given by the formula:

LSA = 2πrh

where LSA is the lateral surface area, π is a mathematical constant approximately equal to 3.14, r is the radius of the cylinder, and h is the height of the cylinder.

Q: How do I find the radius and height of the cylinder?

A: To find the radius and height of the cylinder, you can use the lateral surface area equation:

2πrh = 4224

You can then divide both sides of the equation by 2π to get:

rh = 4224 / (2π)

rh = 4224 / (2 × 3.14)

rh = 4224 / 6.28

rh = 672

Q: What is the relationship between the radius and height of the cylinder?

A: The relationship between the radius and height of the cylinder is given by the equation:

r × h = 672

Q: How do I find the perimeter of the rectangular sheet?

A: To find the perimeter of the rectangular sheet, you can use the formula:

P = 2(l + w)

where P is the perimeter, l is the length, and w is the width.

Q: What is the length of the rectangular sheet?

A: The length of the rectangular sheet is equal to the height of the cylinder (h).

Q: How do I find the height of the cylinder?

A: To find the height of the cylinder, you can use the relationship between the radius and height:

h = 672 / r

Q: What is the radius of the cylinder?

A: The radius of the cylinder is given by the equation:

r = 4224 / (2π × 672)

r = 4224 / (2 × 3.14 × 672)

r = 4224 / 4223.68

r = 1

Q: What is the perimeter of the rectangular sheet?

A: The perimeter of the rectangular sheet is given by the formula:

P = 2(l + w)

P = 2(672 + 33)

P = 2(705)

P = 1410

Q: What is the significance of the lateral surface area of a hollow cylinder?

A: The lateral surface area of a hollow cylinder is an important concept in mathematics and engineering. It is used to calculate the surface area of a cylinder, which is essential in designing and building structures such as pipes, tubes, and cylinders.

Q: How do I apply the concept of lateral surface area to real-world problems?

A: The concept of lateral surface area can be applied to real-world problems in various fields such as engineering, architecture, and design. For example, you can use the lateral surface area to calculate the surface area of a pipe or a tube, which is essential in designing and building structures.

Q: What are some common applications of the lateral surface area of a hollow cylinder?

A: Some common applications of the lateral surface area of a hollow cylinder include:

Calculating the surface area of a pipe or a tube
Designing and building structures such as pipes, tubes, and cylinders
Calculating the surface area of a cylinder in engineering and architecture
Solving problems related to the surface area of a cylinder in mathematics and science

Q: What are some common mistakes to avoid when calculating the lateral surface area of a hollow cylinder?

A: Some common mistakes to avoid when calculating the lateral surface area of a hollow cylinder include:

Not using the correct formula for the lateral surface area
Not substituting the correct values for the radius and height of the cylinder
Not simplifying the equation correctly
Not checking the units of the answer

Q: How do I check my answer for the lateral surface area of a hollow cylinder?

A: To check your answer for the lateral surface area of a hollow cylinder, you can use the following steps:

Plug in the values for the radius and height of the cylinder into the formula for the lateral surface area
Simplify the equation correctly
Check the units of the answer
Compare your answer with the correct answer

Q: What are some common resources for learning about the lateral surface area of a hollow cylinder?

A: Some common resources for learning about the lateral surface area of a hollow cylinder include:

Textbooks on mathematics and science
Online resources such as Khan Academy and Mathway
Calculators and software such as Wolfram Alpha and Mathematica
Online communities and forums such as Reddit and Stack Exchange

The final answer is: 1410