103 LearnersLast updated on July 20th, 2025
Volume of Hollow Sphere
The volume of a hollow sphere is the space it occupies, which is the difference between the volume of the outer sphere and the inner sphere. A hollow sphere consists of two concentric spheres with different radii. To find the volume of a hollow sphere, you subtract the volume of the inner sphere from the volume of the outer sphere. In real life, kids relate to the volume of hollow spheres by thinking of things like a basketball or a spherical shell. In this topic, let’s learn about the volume of a hollow sphere.
What is the volume of a hollow sphere?
The volume of a hollow sphere is the amount of space it occupies.
It is calculated using the formula: Volume = (4/3)π(R³ - r³) Where 'R' is the radius of the outer sphere and 'r' is the radius of the inner sphere.
Volume of Hollow Sphere Formula A hollow sphere is a 3-dimensional shape with two concentric spheres.
To calculate its volume, you subtract the volume of the inner sphere from that of the outer sphere.
The formula for the volume of a hollow sphere is given as follows: Volume = (4/3)π(R³ - r³)
How to Derive the Volume of a Hollow Sphere?
To derive the volume of a hollow sphere, we use the concept of volume as the total space occupied by a 3D object.
A hollow sphere consists of an outer sphere and an inner sphere, both concentric.
Its volume can be derived as follows: The formula for the volume of a sphere is: Volume = (4/3)πr³
For a hollow sphere: Outer volume = (4/3)πR³ Inner volume = (4/3)πr³
The volume of the hollow sphere will be, Volume = Outer volume - Inner volume Volume = (4/3)π(R³ - r³)
How to find the volume of a hollow sphere?
The volume of a hollow sphere is always expressed in cubic units, for example, cubic centimeters (cm³), cubic meters (m³).
Subtract the volume of the inner sphere from the outer sphere to find the volume.
Let’s take a look at the formula for finding the volume of a hollow sphere: Write down the formula Volume = (4/3)π(R³ - r³) 'R' is the radius of the outer sphere, and 'r' is the radius of the inner sphere.
Measure the radii of both spheres. Once you know the radii, substitute those values into the formula Volume = (4/3)π(R³ - r³).
Calculate the volumes of both spheres separately, then subtract to find the hollow sphere's volume.
Tips and Tricks for Calculating the Volume of a Hollow Sphere
Remember the formula: The formula for the volume of a hollow sphere is simple: Volume = (4/3)π(R³ - r³)
Break it down: The volume is the space between the two spheres.
Calculate the outer sphere volume and the inner sphere volume separately before subtracting.
Simplify the numbers: If the radii are simple numbers like 2, 3, or 4, it is easy to cube.
For example, for R = 3 and r = 2, 3³ = 27 and 2³ = 8.
Check for subtraction errors: Ensure you subtract the inner volume from the outer volume.
Common Mistakes and How to Avoid Them in Volume of Hollow Sphere
Making mistakes while learning the volume of the hollow sphere is common.
Let’s look at some common mistakes and how to avoid them to get a better understanding of the volume of hollow spheres.
Mistake 1
Confusing Volume with Surface Area
Some students confuse the formula for volume with the formula for surface area.
Surface area is calculated differently, while volume is the difference in volumes of the two spheres.
For example, the volume is (4/3)π(R³ - r³), not the surface area formula.
Mistake 2
Confusing Volume with Radius
Some kids may think of the sphere’s radius instead of the volume formula.
Volume is the space inside the hollow sphere, whereas radius is a linear measurement. Do not mix them up.
Mistake 3
Using the wrong Formula for a solid sphere
Some kids use the formula for the volume of a solid sphere ((4/3)πr³) instead of the hollow sphere formula.
Mistake 4
Ignoring the difference between the inner and outer spheres
Thinking of volume without considering the hollow nature. This happens when someone uses the outer radius but ignores the inner radius.
Mistake 5
Incorrectly calculating the radii
Some students calculate the given volume without using the correct radii values.
Ensure both radii are accurate for the formula.
Volume of Hollow Sphere Examples
Problem 1
A hollow sphere has an outer radius of 5 cm and an inner radius of 3 cm. What is its volume?
The volume of the hollow sphere is approximately 305.36 cm³.
Explanation
To find the volume of a hollow sphere, use the formula: V = (4/3)π(R³ - r³) Here, the outer radius R is 5 cm, and the inner radius r is 3 cm, so: V = (4/3)π(5³ - 3³) = (4/3)π(125 - 27) = (4/3)π(98) ≈ 305.36 cm³
Problem 2
A hollow sphere has an outer radius of 10 m and an inner radius of 7 m. Find its volume.
The volume of the hollow sphere is approximately 1221.73 m³.
Explanation
To find the volume of a hollow sphere, use the formula: V = (4/3)π(R³ - r³)
Substitute the outer radius (10 m) and inner radius (7 m): V = (4/3)π(10³ - 7³) = (4/3)π(1000 - 343) = (4/3)π(657) ≈ 1221.73 m³
Problem 3
The volume of a hollow sphere is given as approximately 904.32 cm³. If the outer radius is 9 cm, what is the inner radius?
The inner radius of the hollow sphere is approximately 7 cm.
Explanation
If you know the volume of the hollow sphere and need to find the inner radius, rearrange the formula and solve:
V = (4/3)π(R³ - r³) 904.32 ≈ (4/3)π(9³ - r³) Solve for r³: r³ ≈ 9³ - (904.32 × 3)/(4π) r³ ≈ 729 - 215.04 r³ ≈ 513.96
Taking the cube root, r ≈ 7 cm
Problem 4
A hollow sphere has an outer radius of 8 inches and an inner radius of 5 inches. Find its volume.
The volume of the hollow sphere is approximately 1291.99 inches³.
Explanation
Using the formula for volume: V = (4/3)π(R³ - r³) Substitute the outer radius 8 inches and inner radius 5 inches: V = (4/3)π(8³ - 5³) = (4/3)π(512 - 125) = (4/3)π(387) ≈ 1291.99 inches³
Problem 5
You have a hollow spherical shell with an outer radius of 6 feet and an inner radius of 4 feet. How much space (in cubic feet) is available inside the shell?
The hollow shell has a volume of approximately 402.12 cubic feet.
Explanation
Using the formula for volume: V = (4/3)π(R³ - r³) Substitute the outer radius 6 feet and inner radius 4 feet: V = (4/3)π(6³ - 4³) = (4/3)π(216 - 64) = (4/3)π(152) ≈ 402.12 ft³
FAQs on Volume of Hollow Sphere
1.Is the volume of a hollow sphere the same as the surface area?
2.How do you find the volume if the radii are given?
3.What if I have the volume and need to find one of the radii?
4.Can the radii of a hollow sphere be decimals or fractions?
5.What is the difference between a solid sphere and a hollow sphere in terms of volume?
Important Glossaries for Volume of Hollow Sphere
- Outer Radius (R): The radius of the outer sphere in a hollow sphere, measured from the center to the outer surface.
- Inner Radius (r): The radius of the inner sphere in a hollow sphere, measured from the center to the inner surface.
- Volume: The amount of space enclosed within a 3D object. In the case of a hollow sphere, the volume is the difference between the outer and inner volumes, expressed in cubic units.
- Cubic Units: The units of measurement used for volume. If the radii are in centimeters (cm), the volume will be in cubic centimeters (cm³); if in meters, it will be in cubic meters (m³).
- Concentric: Describes two or more objects sharing the same center or axis, as seen in the inner and outer spheres of a hollow sphere.
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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.
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