101 LearnersLast updated on July 20th, 2025
Volume of Hollow Cylinder
The volume of a hollow cylinder is the amount of space it occupies or the capacity it can hold. A hollow cylinder is a 3D shape consisting of two concentric cylinders, one inside the other. To find the volume of a hollow cylinder, we calculate the difference between the volumes of the outer and inner cylinders. In real life, kids encounter hollow cylinders in objects like pipes or tubes. In this topic, let’s learn about the volume of a hollow cylinder.
What is the volume of a hollow cylinder?
The volume of a hollow cylinder is the space enclosed between its outer and inner surfaces.
It is calculated using the formula: Volume = π × (R² - r²) × h Where ‘R’ is the outer radius, ‘r’ is the inner radius, and ‘h’ is the height of the cylinder.
Volume of Hollow Cylinder Formula A hollow cylinder is composed of two cylindrical surfaces with different radii.
To calculate its volume, subtract the volume of the inner cylinder from the volume of the outer cylinder.
The formula for the volume of a hollow cylinder is given as follows: Volume = π × (R² - r²) × h
How to Derive the Volume of a Hollow Cylinder?
To derive the volume of a hollow cylinder, we use the concept of volume as the space enclosed by a 3D object.
The volume is the difference between the volume of the outer cylinder and the inner cylinder.
The formula for the volume of any cylinder is: Volume = π × radius² × height For a hollow cylinder: Outer Cylinder Volume = π × R² × h Inner Cylinder Volume = π × r² × h The volume of the hollow cylinder will be, Volume = π × (R² - r²) × h
How to find the volume of a hollow cylinder?
The volume of a hollow cylinder is expressed in cubic units, such as cubic centimeters (cm³) or cubic meters (m³).
To find the volume, identify the outer and inner radii and the height of the cylinder, and then apply the formula.
Let’s take a look at the steps for finding the volume of a hollow cylinder: Write down the formula: Volume = π × (R² - r²) × h The radii are the distances from the center to the outer and inner surfaces.
Substitute the values of R, r, and h in the formula. Calculate (R² - r²) and then multiply by π and the height.
Tips and Tricks for Calculating the Volume of a Hollow Cylinder
Remember the formula: The formula for the volume of a hollow cylinder is: Volume = π × (R² - r²) × h Break it down: Calculate the area difference of the cross-sections first, then multiply by the height.
Simplify the numbers: If R and r are easy to square, simplify the calculations by finding their squares first.
Check for radius difference: Ensure you correctly calculate the square difference (R² - r²) for accuracy.
Common Mistakes and How to Avoid Them in Volume of Hollow Cylinder
Making mistakes while learning the volume of a hollow cylinder is common.
Let’s look at some common mistakes and how to avoid them to get a better understanding of the volume of hollow cylinders.
Mistake 1
Confusing Volume with Surface Area
Some students confuse the formula for volume with the formula for surface area.
Surface area involves adding the areas of the outer and inner surfaces, while volume is calculated by the difference of squared radii multiplied by π and the height.
Mistake 2
Confusing Volume with Length or Circumference
Some may think of the cylinder's height or circumference instead of the volume formula.
Volume measures the space within the hollow cylinder, not the length or circumference.
Mistake 3
Using the wrong Formula for Solid Cylinders
Some students use the formula for the volume of a solid cylinder (π × radius² × height) instead of the hollow cylinder formula.
Mistake 4
Incorrectly Handling Radii
Errors occur when students mix up the outer and inner radii or forget to square them before subtracting.
Mistake 5
Forgetting to Multiply by π
Some students forget to include π in their calculations, which is crucial for finding the correct volume.
Volume of Hollow Cylinder Examples
Problem 1
A hollow cylinder has an outer radius of 5 cm, an inner radius of 3 cm, and a height of 10 cm. What is its volume?
The volume of the hollow cylinder is 251.33 cm³.
Explanation
To find the volume of a hollow cylinder, use the formula: V = π × (R² - r²) × h Here, R = 5 cm, r = 3 cm, and h = 10 cm, so: V = π × (5² - 3²) × 10 = π × (25 - 9) × 10 = π × 16 × 10 = 160π ≈ 502.65 cm³
Problem 2
A hollow cylinder has an outer radius of 7 m, an inner radius of 5 m, and a height of 12 m. Find its volume.
The volume of the hollow cylinder is 754.76 m³.
Explanation
To find the volume of a hollow cylinder, use the formula: V = π × (R² - r²) × h
Substitute R = 7 m, r = 5 m, h = 12 m: V = π × (7² - 5²) × 12 = π × (49 - 25) × 12 = π × 24 × 12 = 288π ≈ 904.32 m³
Problem 3
The volume of a hollow cylinder is 628 cm³. Its height is 8 cm, and the inner radius is 2 cm. What is the outer radius?
The outer radius of the hollow cylinder is 4 cm.
Explanation
The formula for volume is: V = π × (R² - r²) × h
Given V = 628 cm³, r = 2 cm, h = 8 cm, solve for R: 628 = π × (R² - 2²) × 8 628 = π × (R² - 4) × 8
Divide by π and 8: R² - 4 = 25 R² = 29 R = √29 R ≈ 5.39 cm
Problem 4
A hollow cylinder has an outer radius of 4.5 inches, an inner radius of 3 inches, and a height of 15 inches. What is its volume?
The volume of the hollow cylinder is 472.5 inches³.
Explanation
Using the formula for volume: V = π × (R² - r²) × h
Substitute R = 4.5 inches, r = 3 inches, h = 15 inches: V = π × (4.5² - 3²) × 15 = π × (20.25 - 9) × 15 = π × 11.25 × 15 = 168.75π ≈ 530.14 inches³
Problem 5
You have a hollow cylinder with an outer radius of 6 feet, an inner radius of 4 feet, and a height of 20 feet. How much space (in cubic feet) is available inside the cylinder?
The cylinder has a volume of 1256.64 cubic feet.
Explanation
Using the formula for volume: V = π × (R² - r²) × h Substitute R = 6 feet, r = 4 feet, h = 20 feet: V = π × (6² - 4²) × 20 = π × (36 - 16) × 20 = π × 20 × 20 = 400π ≈ 1256.64 ft³
FAQs on Volume of Hollow Cylinder
1.Is the volume of a hollow cylinder the same as the surface area?
2.How do you find the volume if the radii and height are given?
3.What if I have the volume and need to find the outer radius?
4.Can the radii be decimal or fractional values?
Important Glossaries for Volume of Hollow Cylinder
- Outer Radius (R): The distance from the center to the outer surface of the hollow cylinder.
- Inner Radius (r): The distance from the center to the inner surface of the hollow cylinder.
- Height (h): The distance between the cylinder's bases.
- Volume: The amount of space enclosed within a hollow cylinder, calculated by the formula V = π × (R² - r²) × h.
- Cubic Units: The units of measurement used for volume, such as cubic centimeters (cm³) or cubic meters (m³).
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Seyed Ali Fathima S
About the Author
Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.
Fun Fact
: She has songs for each table which helps her to remember the tables
