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Volume of a Hollow Cylinder

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A cylinder is a three-dimensional object that is formed when a rectangle is rotated along any of its sides. A hollow cylinder is one type of cylinder that is hollow from the inside. A hollow cylinder can be defined as a three-dimensional geometric object that is empty from the inside. A hollow cylinder consists of two circular bases that have inner and outer radii. The circular bases are similar to an annular ring, which is a region bounded by two concentric circles.

Hollow Cylinder

A cylinder, which is empty from the inside and has some difference between the internal and external radius is called as H. We can see some thickness enclosed between the inner radius and the outer radius, which is equal to the difference between the internal and external radius. The height, or altitude, of the hollow cylinder, is the perpendicular distance between its two circular bases. Straws, water pipes, tubes, toilet paper rolls, etc. are some examples of hollow cylinders that we see in our daily lives.

Hollow Cylinder

Volume of Hollow Cylinder Formula

The three-dimensional space enclosed by a hollow cylinder is referred to as its volume. For example, the maximum space that can be occupied by oil when the oil is poured into a cylindrical glass jar is the volume of the jar. Now, the formula to calculate the volume of a hollow cylinder is given as follows:

Volume of Hollow Cylinder = π(R2 -r2)h cubic units

where,
h” is the height of the hollow cylinder,
R” is the outer radius of the given cylinder, and
r” is the inner radius of the given cylinder.

Volume of Hollow Cylinder Derivation

Derivation of Volume of Hollow Cylinder

We know that the formula to calculate the volume of a solid cylinder when radius (r) and height (h) is (Ï€r2)h cubic units.

Volume of cylinder = Base area × Height = (πr2)h cubic units

Now, the volume of a hollow cylinder is equal to the difference between the volume of the external cylinder and the volume of the internal cylinder. 

Let us consider that the outer radius of the hollow cylinder is “R,” its inner radius is “r,” and “h” is the height of the cylinder.

Volume of a hollow cylinder = (Volume of the cylinder with outer radius “R” and height “h”) – (Volume of the cylinder with inner radius “r” and height “h”)

Volume of a hollow cylinder (V) = πR2 × h – πr2 × h

V = π(R2 -r2)h cubic units

How to Find the Volume of Hollow Cylinder?

Let’s take an example to understand how to calculate the volume of a hollow cylinder.

Example: Calculate the volume of a hollow cylinder whose external radius is 12 cm, the internal radius is 9 cm, and the height is 7 cm

Solution:

Step 1: Note the values of the given dimensions. Here, the external radius (R) is 12 cm, the internal radius (r) is 9 cm, and the height (h) is 7 cm.

Step 2: We know that the formula to find the volume of a hollow cylinder is [π × (R2−r2) × h] cubic units. Now, substitute the given values in the formula.

Step 3: Thus, the volume of a hollow cylinder is calculated as

V = (22/7) × (122 – 92) × 7
V = 1,386 cm3

Solved Examples on Volume of Hollow Cylinder

Example 1: Calculate the volume of a hollow cylinder whose external radius is 10 cm, the internal radius is 5 cm, and the height is 8 cm [Use π= 22/7]

Solution:

External Radius (R) = 10 cm

Internal Radius (r) = 5 cm

Height (h) = 8 cm

We know that,

The volume of a hollow cylinder = π × (R2−r2) × h cubic units
                                                   = (22/7) × (102 – 52) × 8
                                                   = (22/7) × 75 × 8 
                                                   = 1,885.714 cm3

Hence, the volume of the given hollow cylinder is 1,885.714 cm3

Example 2: Calculate the volume of a hollow cylinder whose external radius is 12 inches, the internal radius is 8 inches, and the height is 6 inches. [Use π= 22/7]

Solution:

External Radius (R) = 12 inches

Internal Radius (r) = 8 inches

Height (h) = 6 inches

We know that,

The volume of a hollow cylinder = π × (R2 -r2) × h cubic units
                                                   = (22/7) × (122 – 82) × 6
                                                   = (22/7) × 80 × 6 
                                                   = 1,508.571 in3

Hence, the volume of the given hollow cylinder is 1,508.571 in3

Example 3: Determine the volume of a hollow cylindrical tube whose thickness is 4 m, the internal radius is 5 m, and the height is 8 m. [Use π= 22/7]

Solution:

Internal Radius (r) = 5 m

The thickness of the cylindrical tube = 4 m

t = R – r = 4 m

R – 5 = 4 

R = 4 + 5 = 9 m

Height (h) = 8 m

We know that,

The volume of a hollow cylinder = π × (R2 -r2) × h cubic units
                                                   = (22/7) × (92 – 52) × 8
                                                   = (22/7) × 56 × 6 = 1,408 m3

Hence, the volume of the given hollow cylinder is 1,408 m3

Example 4: Calculate the height of a hollow cylinder whose outer diameter is 16 cm, the inner diameter is 12 cm, and the volume is 792 cm3. [Use π= 22/7]

Solution:

Outer diameter = 16 cm

So, External Radius (R) = 16 cm/ 2 = 8 cm

Inner diameter = 12 cm

So, Internal Radius (r) = 12 cm/2 = 6 cm

Height = ?

We know that,

Volume of a hollow cylinder = π × (R2 -r2) × h cubic units
                                      792 = (22/7) × (82 – 62) × H
                                        H  = 9

Hence, the height of hollow cylinder is 9 cm

FAQs on Volume of Hollow Cylinder

Question 1: What is a hollow cylinder? Give some examples of a hollow cylinder.

Answer:

A hollow cylinder can be defined as a three-dimensional geometric object that is empty from the inside. A hollow cylinder consists of two circular bases that have inner and outer radii. Straws, water pipes, tubes, toilet paper rolls, etc. are some examples of hollow cylinders that we see in our daily lives.

Question 2: What is the formula to calculate the thickness of a hollow cylinder?

Answer:

The thickness of a hollow cylinder is the enclosed space between the inner radius and the outer radius, which is equal to the difference between the internal and external radius.

Thickness of the hollow cylinder (t) = R − r

where,
R” is the outer radius of the given cylinder
r” is the inner radius of the given cylinder

Question 3: What is the formula to calculate the total surface area of a hollow cylinder?

Answer:

The total surface area of the hollow cylinder is the sum of its curved surface area and the areas of its two circular bases.

Total Surface Area of Hollow Cylinder = [2Ï€h (R + r) + 2Ï€(R² – r²)] square units

where,
h” is the height of the hollow cylinder,
R” is the outer radius of the given cylinder, and
r” is the inner radius of the given cylinder.

Question 4: What does the volume of a hollow cylinder change as its height triples?

Answer:

From the formula for the volume of a hollow cylinder, we can conclude that the volume is directly proportional to the height of the hollow cylinder. So, as the height of the hollow cylinder triples, its volume will also be tripled.

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Last Updated : 30 Jan, 2024
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