A hollow cylinder is a fascinating figure in the world of geometry. Unlike other solid figures like cubes, cones, and regular cylinders, a hollow cylinder is essentially a cylinder that is void on the inside with a certain thickness on the outside. In this article, we will delve into calculating the volume of a hollow cylinder, exploring the formula, its derivation, and some practical examples.

Volume of a Hollow Cylinder Formula 

A hollow cylinder is essentially a cylinder that is empty from the inside with a discernible difference between the internal and external radius. The base of a hollow cylinder resembles an annular ring, or in simpler terms, it looks like a region bounded by two concentric circles .

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The volume of a hollow cylinder can be calculated if we know the outer and inner radius along with the height. The formula is as follows:

Volume of Hollow cylinder, V = π (R 2 – r 2 ) h cubic units

Here, “R” represents the outer radius, “r” is the inner radius, and “h” stands for the height of the hollow cylinder.

The volume of a regular cylinder is calculated as the Base area multiplied by Height = (πr 2 ) × h cubic units, where “r” is the radius and “h” is the height of the cylinder.

To calculate the volume of a hollow cylinder, we subtract the volume of the internal cylinder from the volume of the external cylinder. This gives us the formula for the volume of a hollow cylinder with outer radius “R”, inner radius “r”, and height “h” as follows:

Volume of Hollow cylinder = Volume of a cylinder with radius “R” and height “h” – Volume of a cylinder with radius “r” and height “h”.

This simplifies to: Volume of hollow cylinder = πR 2 × h – πr 2 × h

Which further simplifies to: V = π (R 2 -r 2 )h cubic units

So, the derived formula for the volume of the hollow cylinder, V = π (R 2 -r 2 )h cubic units.

How to Find the Volume of a Hollow Cylinder

To find the volume of a hollow cylinder, we need to know three things:

• The outer radius (R)
   
• The inner radius (r)
   
• The height (h)

The hollow cylinder is like a tube — the space inside the walls is what we calculate as volume.

Steps to Calculate the Volume

Step 1:
Write down the given values for outer radius (R), inner radius (r), and height (h).
Make sure all measurements are in the same unit (like cm, m, etc.).

Step 2:
Use the formula:

Volume=π(R2 −r2)×h

Where:

• π is a constant (Use 22/7 or 3.1416)
   
• R is the outer radius
   
• r is the inner radius
   
• h is the height of the cylinder

Step 3:
Do the calculation and write the answer with proper units (like cm³, m³, or units³).

Applications of Volume of a Hollow Cylinder

The formula for the volume of a hollow cylinder is widely used in both real-life and industrial applications. Since a hollow cylinder has a space between two concentric circular surfaces, knowing its volume is helpful in many practical situations. Below are some key applications:

1. Pipes and Tubes

Hollow cylinders are commonly used to design water pipes, gas lines, and drainage systems. Engineers calculate the volume to determine how much fluid or gas a pipe can hold or how much material is needed to manufacture the pipe.

2. Storage Tanks

Large cylindrical tanks used to store oil, fuel, chemicals, or water are often hollow cylinders. Calculating their volume helps determine how much substance they can safely store.

3. Construction and Architecture

In building design, hollow cylindrical columns are used for support structures. Knowing the volume helps estimate material requirements (like concrete or steel) and weight-bearing capacity.

4. Mechanical Engineering

Hollow cylinders are used in shafts, rollers, and engine components to reduce weight while maintaining strength. Volume calculation is important for material efficiency and performance.

5. Medical Devices

Syringes, catheters, and many medical instruments have hollow cylindrical shapes. Calculating volume ensures accurate delivery of fluids or medicines.

6. Space and Aerospace Engineering

Futuristic designs like space stations or sci-fi structures (e.g., Rama in Encounter with Rama) are often imagined as hollow cylinders due to their large usable space. Calculating the volume helps with space planning and human habitation design.

7. Manufacturing

When making hollow cylindrical products (e.g., metal pipes, plastic tubes, cardboard rolls), industries need volume measurements to estimate material costs and production waste.

Examples on Volume of a Hollow Cylinder

 Find the volume of a hollow cylinder with an outer radius of 10 cm, inner radius of 6 cm, and height of 14 cm.
(Take π = 22/7)

Solution:
Volume = (22/7) × (10² − 6²) × 14
    = (22/7) × (100 − 36) × 14
    = (22/7) × 64 × 14
    = (22 × 896) / 7
    = 19712 / 7
    = 2816 cm³

Example 2:

Q2. The height of a hollow cylinder is 20 cm. If the outer radius is 12 cm and the inner radius is 8 cm, find its volume. (Use π = 3.14)

Solution:
Volume = 3.14 × (12² − 8²) × 20
    = 3.14 × (144 − 64) × 20
    = 3.14 × 80 × 20
    = 3.14 × 1600
    = 5024 cm³

Example 3:

Q3. A pipe is in the shape of a hollow cylinder with height 35 cm. If the outer and inner radii are 7 cm and 5 cm respectively, find its volume.
(Take π = 22/7)

Solution:
Volume = (22/7) × (7² − 5²) × 35
    = (22/7) × (49 − 25) × 35
    = (22/7) × 24 × 35
    = (22 × 840) / 7
    = 18480 / 7
    = 2640 cm³

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