109 LearnersLast updated on July 9th, 2025
Volume of Hollow Cone
The volume of a hollow cone refers to the space contained within its thin, shell-like structure. A hollow cone is a 3D shape with a circular base and a vertex, resembling a regular cone but lacking the solid interior. To find the volume of a hollow cone, we subtract the volume of the smaller cone (removed portion) from the larger cone. In real life, hollow cones can be found in structures like traffic cones, funnels, or megaphones. In this topic, let’s learn about the volume of a hollow cone.
What is the volume of the hollow cone?
The volume of a hollow cone is the space it encloses, calculated by subtracting the volume of the smaller cone from the larger cone.
The formula is: Volume = (1/3)πh(R² - r²) Where ‘R’ is the base radius of the larger cone, ‘r’ is the base radius of the smaller cone, and ‘h’ is the height of the cone.
This formula accounts for the conical shape with a missing interior section.
How to Derive the Volume of a Hollow Cone?
To derive the volume of a hollow cone, we begin with the volume of a solid cone and subtract the volume of the smaller cone that creates the hollow space.
The formula for the volume of a cone is: Volume = (1/3)πr²h
For a hollow cone: The volume of the larger cone minus the volume of the smaller cone gives: Volume = (1/3)πh(R² - r²)
How to find the volume of a hollow cone?
The volume of a hollow cone is expressed in cubic units, such as cubic centimeters (cm³) or cubic meters (m³).
To find it, identify the radii of both cones and the height.
Here’s how: Write down the formula: Volume = (1/3)πh(R² - r²)
- Identify the radii of the two cones and the height.
- Substitute these values into the formula.
- This calculation will give you the volume of the hollow section, subtracting the smaller cone’s volume from the larger cone’s volume.
Tips and Tricks for Calculating the Volume of Hollow Cone
- Remember the formula: Volume = (1/3)πh(R² - r²)
- Break it down: Calculate the volume of each cone separately and subtract to find the hollow space.
- Simplify the numbers: If R and r are simple numbers, it’s easy to compute R² and r².
- Check for accuracy: Ensure you use the correct radius for each cone and maintain consistent height measurements.
Common Mistakes and How to Avoid Them in Volume of Hollow Cone
Mistakes are common while learning about the volume of a hollow cone. Let’s explore some common errors and how to avoid them for a clearer understanding.
Mistake 1
Confusing Radius with Diameter
Students often confuse radius with diameter. Remember that the radius is half the diameter. Use the radius in the formula, not the diameter.
Mistake 2
Mixing Up the Radii of the Cones
Ensure you correctly identify which radius belongs to the larger cone and which to the smaller one. Mixing them up will lead to incorrect calculations.
Mistake 3
Forgetting to Subtract Volumes
Some students forget to subtract the volume of the smaller cone from the larger cone, which results in calculating the volume of a solid cone instead of a hollow one.
Mistake 4
Incorrect Formula Application
Using the formula for a solid cone instead of the hollow cone results in errors. Ensure you apply the correct formula, which accounts for the hollow space.
Mistake 5
Inconsistent Units
Mixing units (e.g., using centimeters for radius and meters for height) leads to errors. Ensure all measurements are in the same unit before calculating.
Volume of Hollow Cone Examples
Problem 1
A hollow cone has an external radius of 5 cm, an internal radius of 3 cm, and a height of 10 cm. What is its volume?
The volume of the hollow cone is 209.44 cm³.
Explanation
To find the volume of a hollow cone, use the formula: V = (1/3)πh(R² - r²)
Here, R = 5 cm, r = 3 cm, and h = 10 cm,
so: V = (1/3)π(10)(5² - 3²) = (1/3)π(10)(25 - 9) = (1/3)π(10)(16) = (1/3)π(160) ≈ 209.44 cm³
Problem 2
Calculate the volume of a hollow cone with an external radius of 8 m, an internal radius of 6 m, and a height of 12 m.
The volume of the hollow cone is 904.78 m³.
Explanation
Using the formula: V = (1/3)πh(R² - r²)
Substitute R = 8 m, r = 6 m, and h = 12 m:
V = (1/3)π(12)(8² - 6²) = (1/3)π(12)(64 - 36) = (1/3)π(12)(28) = (1/3)π(336) ≈ 904.78 m³
Problem 3
The volume of a hollow cone is 150 cm³, with an external radius of 7 cm and a height of 9 cm. Find the internal radius.
The internal radius of the cone is approximately 6.49 cm.
Explanation
To find the internal radius,
use the formula: 150 = (1/3)π(9)(7² - r²)
450 = π(9)(49 - r²)
450 = 9π(49 - r²)
50 = π(49 - r²)
50/π = 49 - r²
r² = 49 - (50/π)
r ≈ √6.49 r ≈ 6.49 cm
Problem 4
A hollow cone has a height of 15 inches, an external radius of 7 inches, and an internal radius of 5 inches. Find its volume.
The volume of the hollow cone is approximately 628.32 inches³.
Explanation
Using the formula for volume: V = (1/3)πh(R² - r²)
Substitute R = 7 inches, r = 5 inches, and h = 15 inches:
V = (1/3)π(15)(7² - 5²) = (1/3)π(15)(49 - 25) = (1/3)π(15)(24) = (1/3)π(360) ≈ 628.32 inches³
Problem 5
You have a hollow cone-shaped funnel with an external radius of 10 cm, an internal radius of 8 cm, and a height of 20 cm. How much space is available inside the funnel?
The funnel has a volume of approximately 2513.27 cm³.
Explanation
Using the formula for volume: V = (1/3)πh(R² - r²)
Substitute R = 10 cm, r = 8 cm, and h = 20 cm:
V = (1/3)π(20)(10² - 8²) = (1/3)π(20)(100 - 64) = (1/3)π(20)(36) = (1/3)π(720) ≈ 2513.27 cm³
FAQs on Volume of Hollow Cone
1.Is the volume of a hollow cone the same as a solid cone?
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 internal radius?
4.Can the radii be decimals or fractions?
5.What is the significance of the height in the volume formula?
Important Glossaries for Volume of Hollow Cone
- External Radius (R): The radius of the base of the larger cone in a hollow cone.
- Internal Radius (r): The radius of the base of the smaller cone in a hollow cone.
- Height (h): The perpendicular distance from the base to the apex of the cone.
- Volume: The amount of space enclosed within a 3D object, calculated for a hollow cone using the formula (1/3)πh(R² - r²).
- Cubic Units: The units of measurement used for volume, such as cm³ or m³, indicating the space enclosed by the hollow cone.
Explore More measurement
![Important Math Links Icon]()
Previous to Volume of Hollow Cone
![Important Math Links Icon]()
Next to Volume of Hollow Cone
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
