The Volume of a Hollow Cylinder

Reviewed by the Helping with Math Team · Updated 13 October 2026

Introduction

We frequently encounter solid shapes in our daily lives. We utilize, touch, and sense them. Objects with two dimensions: length and breadth are called 2D objects. A shape with three dimensions of length, breadth, and height is referred to as a three-dimensional shape or a solid shape. Solid shapes are also known as 3D shapes. By looking at them from various locations and angles, they appear to be distinct. The hollow cylinder is a 3-dimensional shape. The volume of the hollow cylinder is an important formula to be learned.

In this article first, we will learn what is a cylinder. Then we will understand what a hollow cylinder is. Next, we will study what volume is. Then, find formulas for the volume of a cylinder. Finally, we comprehensively discuss different formulas to calculate the volume of a hollow cylinder using examples.

What is a cylinder?

The cylinder is a three-dimensional shape with two parallel, round surfaces. These congruent circular surfaces could be either circles or ellipses. A curved side is also present. The axis is the line that joins the centers of the two bases. The circumference of the base makes up the width of the cylinder’s side, while the height of the cylinder makes up its length.

A cylinder has 2 bases, 3 faces (2 flat faces and 1 curved face), 2 edges and 0 vertices.

What are the typical examples of cylinder shapes?

Candle, Beaker, Water Tanks, Fire Extinguisher, Cold Drink Cans, Gas Cylinder, Battery, Pencil Holder, Cell, Oil Tank, Pencil, Dustbin, Log of Wood, Perfume/ Deodorant Bottle, Industrial Chimney, Drum, Cake

• A cake
• Drink cans
• Gas cylinder
• Fire extinguisher
• Candles
• A log
• Chimneys
• A drum

Properties of Cylinder

• The two bases (flat faces) are parallel and congruent circles.
• It has two curved edges, where the bases meet the side.
• It has one curved face.
• It has no vertices (corners).
• The axis is the line joining the centers of the two bases.
• The cylinder’s total surface area is made up of its bases and sides.
• Space inside a cylinder makes up its volume.

Cylinder components

Base and Sides

A cylinder has two circular bases joined by a curved side. If you could ‘unroll’ the side, it would straighten out into a rectangle.

Height

Height h is the distance between the two bases perpendicularly. It is essential to use the perpendicular height (sometimes known as “altitude”) even when a cylinder is tilted.

Radius

The radius r of a cylinder is the same as the base’s radius. Half of the diameter can be used to compute the radius.

Example:

Complete the below table.

Solution:

Now we will learn what is a hollow cylinder.

What is a hollow cylinder?

A solid cylinder is like a stack of round disks. A hollow cylinder is a cylinder with a smaller cylinder removed from its middle, like a pipe. It has an outer (external) radius R and an inner (internal) radius r, and the difference R − r is the thickness of its wall.

What are the typical examples of hollow cylinder shapes?

Toilet paper roll, Water well, circular stadium, straws, pipes, bearing bushes

• Paper roll
• A well
• Round stadium
• Straws
• Bearing bushes

What is volume?

Volume is the amount of space an object takes up. Knowing it tells us how much is needed to fill something, such as the water in a bottle, aquarium or tank. Volume is measured in cubic units.

Cubic centimeters: cm³

Cubic inches: in³

Cubic feet: ft³

Cubic meters: m³

What is the volume of a cylinder?

The volume of a cylinder is how much space it fills, or how much it can hold. The base area is the area of one round base, and the volume is the base area multiplied by the height.

The volume of a cylinder=Base area of the cylinder × Height of the cylinder

Example:

Find the volume of the following cylinders.

1. Cylinder A with a base area of 14 cm² and a height of 10 cm
2. Cylinder B with a base area of 7 m² and a height of 10 m
3. Cylinder C with a base area of 21 ft² and a height of 10 ft

Solution:

1. The volume of cylinder A = Base area of cylinder A x Height of cylinder A=14 cm²×10 cm=140 cm³
2. The volume of cylinder B=Base area of cylinder B x Height of cylinder B=7 m²×10 m=70 m³
3. The volume of cylinder C=Base area of cylinder C x Height of cylinder C=21 ft²×10 ft=210 ft³

If the base area of a cylinder is not given, we have to first calculate the base area of the cylinder to find the volume of the cylinder.

The base area of the cylinder could be calculated using the below formula.

The base area of the cylinder=π×radius²

The base area of the cylinder=π×r²

Steps to calculate the above formula.

1. The radius of the base of the cylinder should be calculated.
2. Find the radius’s square.
3. Multiply the radius’s square by π.
4. The answer is given in square units of the length radius.
   • The result for the area is in cm² if the radius is specified in cm.
   • The result for the area is in m² if the radius is specified in m.
   • The result for the area is in ft² if the radius is specified in ft.

Example:

Find the base area of the following cylinders.

1. Cylinder A with a radius of 14 cm
2. Cylinder B with a radius of 7 m
3. Cylinder C with a radius of 21 ft

Solution:

1. The base area of cylinder A=π×(the radius of cylinder A)²=π×(14 cm)²=$\frac{22}{7}$ x (14 cm×14 cm)=616 cm²
2. The base area of cylinder B=π×(the radius of cylinder B)²=π×(7 m)²=$\frac{22}{7}$ x (7 m×7 m)=154 m²
3. The base area of cylinder C=π×(the radius of cylinder C)²=π×(21 ft)²=$\frac{22}{7}$ x (21 ft×21 ft)=1,386 ft²

Now we know that,

The base area of the cylinder=π×radius²

Now, we can expand the formula of the volume of a cylinder.

The volume of a cylinder=Base area of the cylinder × Height of the cylinder=(π×radius²) x Height of the cylinder=π×r² x Height of the cylinder

Example:

Find the volume of the following cylinders.

1. Cylinder A with a radius of 14 cm and a height of 10 cm
2. Cylinder B with a radius of 7 m and a height of 10 m
3. Cylinder C with a radius of 21 ft and a height of 10 ft

Solution:

1. The volume of cylinder A = Base area of cylinder A × Height of cylinder A=π×The radius of cylinder A²×Height of cylinder A=π×(14 cm)²×10 cm = $\frac{22}{7}$ x 14 cm×14 cm×10 cm=6,160 cm³
2. The volume of cylinder B = Base area of cylinder B × Height of cylinder B=π×The radius of cylinder B²×Height of cylinder B=π×(7 m)²×10 m = $\frac{22}{7}$ x 7 m×7 m×10 m=1,540 m³
3. The volume of cylinder C=Base area of cylinder C×Height of cylinder C=π×The radius of cylinder C²×Height of cylinder C=π×(21 ft)²×10 ft = $\frac{22}{7}$ x 21 ft×21 ft×10 ft=13,860 ft³

What is the volume of a hollow cylinder?

The volume of a hollow cylinder could be derived by using the formula for the volume of a cylinder. The volume of a hollow cylinder equals the difference between the volume of the external cylinder and the volume of the internal cylinder.

Formula 1: The volume of a hollow cylinder

The volume of a hollow cylinder=The volume of the external cylinder-The volume of the internal cylinder

In other words, the volume of a hollow cylinder equals the difference between the volume of a solid cylinder and the volume of the inner cylinder.

The volume of a hollow cylinder=The volume of a solid cylinder-The volume of an inner cylinder

The above formula could be graphically presented as follows.

Example:

Find the volume of the following hollow cylinders.

1. Hollow cylinder A

The volume of the solid cylinder is 675 cm³ and the volume of the inner cylinder is 475 cm³

2. Hollow cylinder B

The volume of the solid cylinder is 50 m³ and the volume of the inner cylinder is 35 m³

3. Hollow cylinder C

The volume of the solid cylinder is 100 ft³ and the volume of the inner cylinder is 80 ft³

Solution:

1. The volume of hollow cylinder A=Solid cylinder’s volume of hollow cylinder A-Inner cylinder’s volume of hollow cylinder A=675 cm³-475 cm³=200 cm³
2. The volume of hollow cylinder B=Solid cylinder’s volume of hollow cylinder B-Inner cylinder’s volume of hollow cylinder B=50 m³-35 m³=15 m³
3. The volume of hollow cylinder C=Solid cylinder’s volume of hollow cylinder C-Inner cylinder’s volume of hollow cylinder C=100 ft³-80 ft³=20 ft³

We can determine the volume of the hollow cylinder if we know the base areas of the external (Solid) and interior (Inner) cylinders. We have to find the difference between the area of the external (Solid) cylinder and the internal (Inner) cylinder, and then multiply it by the height of the cylinders.

It can be demonstrated using the formula below.

Formula 2: The volume of a hollow cylinder

The volume of a hollow cylinder=(The base area of the external cylinder − The base area of the internal cylinder)×Cylinder’s height

Example:

Find the volume of the following hollow cylinders.

1. Hollow cylinder A

The base area of the solid cylinder is 675 cm², the base area of the inner cylinder is 475 cm², and the height of the cylinder is 10 cm

2. Hollow cylinder B

The base area of the solid cylinder is 50 m², the base area of the inner cylinder is 25 m², and the height of the cylinder is 10 m

3. Hollow cylinder C

The base area of the solid cylinder is 100 ft², the base area of the inner cylinder is 80 ft², and the height of the cylinder is 10 ft

Solution:

1. The volume of hollow cylinder A=(Solid cylinder’s base area of hollow cylinder A − Inner cylinder’s base area of hollow cylinder A)×height of hollow cylinder A=(675 cm² − 475 cm²)×10 cm=2,000 cm³
2. The volume of hollow cylinder B=(Solid cylinder’s base area of hollow cylinder B − Inner cylinder’s base area of hollow cylinder B)×height of hollow cylinder B=(50 m² − 25 m²)×10 m=250 m³
3. The volume of hollow cylinder C=(Solid cylinder’s base area of hollow cylinder C − Inner cylinder’s base area of hollow cylinder C)×height of hollow cylinder C=(100 ft² − 80 ft²)×10 ft=200 ft³

If the volume or base area of the external cylinder (Solid cylinder) and internal cylinder (inner cylinder) is not given, we have to find the volume of the external cylinder (Solid cylinder) and internal cylinder (inner cylinder) separately to find the volume of a hollow cylinder.

Let’s consider the radius of the external cylinder as “R”, the radius of the internal cylinder as “r” and the height of the hollow cylinder as “h”.

The volume of a hollow cylinder=The volume of the external cylinder-The volume of the internal cylinder

The volume of the external cylinder=The base area of the external cylinder x Height of the cylinder=π×(the radius of the external cylinder)² x Height of the cylinder=π×R²h

The volume of the internal cylinder=The base area of the internal cylinder x Height of the cylinder=π×(the radius of the internal cylinder)² x Height of the cylinder=π×r²h

Therefore,

Formula 3: The volume of a hollow cylinder

The volume of a hollow cylinder=π×R²h-π×r²h

Example:

Find the volume of the following hollow cylinders.

1. Hollow cylinder A

The radius of the external cylinder is 14 cm, the radius of the internal cylinder is 7 cm, and the height of the cylinder is 10 cm

2. Hollow cylinder B

The radius of the external cylinder is 7 m, the radius of the internal cylinder is 3.5 m, and the height of the cylinder is 10 m

3. Hollow cylinder C

The radius of the external cylinder is 21 ft, the radius of the internal cylinder is 7 ft, and the height of the cylinder is 10 ft

Solution:

1. The volume of a hollow cylinder A=π×R²h-π×r²h=$\frac{22}{7}$ x (14 cm)²×10 cm-$\frac{22}{7}$ x (7 cm)²×10 cm=6,160 cm³-1,540 cm³=4,620 cm³
2. The volume of a hollow cylinder B=π×R²h-π×r²h=$\frac{22}{7}$ x (7 m)²×10 m-$\frac{22}{7}$ x (3.5 m)²×10 m=1,540 m³-385 m³=1,155 m³
3. The volume of a hollow cylinder C=π×R²h-π×r²h=$\frac{22}{7}$ x (21 ft)²×10 ft-$\frac{22}{7}$ x (7 ft)²×10 ft=13,860 ft³-1,540 ft³=12,320 ft³

Now, we will try to simplify the below formula and derive another formula to find the volume of a hollow cylinder.

The volume of a hollow cylinder=π×R²h-π×r²h

In the above formula, there are some common terms.

• π (or $\frac{22}{7}$) is a common term for the volume calculation of the external cylinder of a hollow cylinder and the volume calculation of the internal cylinder of a hollow cylinder
• h or height is also a common term for the volume calculation of the external cylinder of a hollow cylinder and the volume calculation of the internal cylinder of a hollow cylinder

Let’s consider π and height as common terms and rearrange the formula of the volume of a hollow cylinder.

I knowthe inner radius rthe thickness tOuter Rr or tHeight h

V = π × 10 × (14² − 7²) = π × 10 × 147 ≈ 4,618.14 cubic units (with π = 22/7: 4,620).

Formula 4: The volume of a hollow cylinder

The volume of a hollow cylinder=π×R²h: π×r²h=π×h×(R²-r²)

Example:

Find the volume of the following hollow cylinders using the below formula.

The volume of a hollow cylinder=π×h×(R²-r²)

1. Hollow cylinder A

The radius of the external cylinder is 14 cm, the radius of the internal cylinder is 7 cm, and the height of the cylinder is 10 cm

2. Hollow cylinder B

The radius of the external cylinder is 7 m, the radius of the internal cylinder is 3.5 m, and the height of the cylinder is 10 m

3. Hollow cylinder C

The radius of the external cylinder is 21 ft, the radius of the internal cylinder is 7 ft, and the height of the cylinder is 10 ft

Solution:

1. The volume of a hollow cylinder A=π×h×(R²-r²)=$\frac{22}{7}$×10 cm×((14 cm)² − (7 cm)²)=$\frac{22}{7}$×10 cm x 147 cm²=4,620 cm³
2. The volume of a hollow cylinder B=π×h×(R²-r²)=$\frac{22}{7}$×10 m×((7 m)² − (3.5 m)²)=$\frac{22}{7}$×10 m x 36.75 m²=1,155 m³
3. The volume of a hollow cylinder C=π×h×(R²-r²)=$\frac{22}{7}$×10 ft×((21 ft)² − (7 ft)²)=$\frac{22}{7}$×10 ft x 392 ft²=12,320 ft³

Let’s try to find the volume of a hollow cylinder when the thickness of the hollow cylinder is given. Let’s consider the radius of the external cylinder as “R”, the radius of the internal cylinder as “r”, the thickness of the hollow cylinder as “t” and the height of the hollow cylinder as “h”.

The thickness of a hollow cylinder=The radius of the external cylinder-The radius of the internal cylinder t=R-r

We can rearrange above the formula as follows.

The radius of the internal cylinder=The radius of the external cylinder-Thickness of a hollow cylinder r=R-t

Now, we can modify the formula for the volume of the hollow cylinder using the above formula.

Formula 5: The volume of a hollow cylinder

The volume of a hollow cylinder=π×h×(R²-r²)=π×h×(R²-(R-t)²)

Example:

Find the volume of the following hollow cylinders.

1. Hollow cylinder A

The radius of the external cylinder is 14 cm, the thickness of the hollow cylinder is 7 cm, and the height of the cylinder is 10 cm

2. Hollow cylinder B

The radius of the external cylinder is 7 m, the thickness of the hollow cylinder is 3.5 m, and the height of the cylinder is 10 m

3. Hollow cylinder C

The radius of the external cylinder is 21 ft, the thickness of the hollow cylinder is 14 ft, and the height of the cylinder is 10 ft

Solution:

1. The volume of a hollow cylinder A=π×h×(R²-(R-t)²)=$\frac{22}{7}$×10 cm×((14 cm)² − (14 cm − 7 cm)²)=$\frac{22}{7}$×10 cm x 147 cm²=4,620 cm³
2. The volume of a hollow cylinder B=π×h×(R²-(R-t)²)=$\frac{22}{7}$×10 m×((7 m)² − (7 m − 3.5 m)²)=$\frac{22}{7}$×10 m x 36.75 m²=1,155 m³
3. The volume of a hollow cylinder C=π×h×(R²-(R-t)²)=$\frac{22}{7}$×10 ft×((21 ft)² − (21 ft − 14 ft)²)=$\frac{22}{7}$×10 ft x 392 ft²=12,320 ft³

Summary

• The cylinder is a three-dimensional shape with two parallel, round surfaces. A cylinder that has some discrepancy between the internal and external radius is called a hollow cylinder. The volume of cylinder calculation method is important in calculating the volume of a hollow cylinder. The volume of a cylinder is calculated by multiplying the base area of the cylinder by its height. The volume of a hollow cylinder equals the difference between the volume of the external cylinder and the volume of the internal cylinder. Multiple formulas can be formed by extending and rearranging the fundamental formula for the volume of a hollow cylinder.
• The volume of a hollow cylinder=The volume of the external cylinder-The volume of the internal cylinder
• The volume of a hollow cylinder=(The base area of the external cylinder − The base area of the internal cylinder)×Cylinder’s height
• The volume of a hollow cylinder=π×R²h-π×r²h
• The volume of a hollow cylinder=π×h×(R²-r²)
• The volume of a hollow cylinder=π×h×R²-(R-t)²

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