Volume of a Cylinder: Formula, Units, Derivation, Surface Areas & Solved Examples

The volume of a cylinder is the three-dimensional space occupied by it or the quantity of material that a cylinder can enclose. The formula for calculating volume of a cylinder in mathematics is equal to \(πr^2\times h\), where r is the radius of the circular base and h is the height of the cylinder.

The value of volume of any material is measured in cubic units. The volume of a cylinder thus measures the number of cubic units that will exactly fill the interior space of the cylinder. In this article we will learn about cylinders, their types and methods of calculating the volume of a cylinder with the help of some important examples.

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What is a Cylinder?

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A cylinder is a solid shape with two circular bases parallel to one another and is linked by a curved surface. This curved surface when visualized as a cylinder cut open is in fact a rectangle. The center of the cylinder has a line passing such that it joins the two circular discs and is termed the cylinder axis.  The volume of the material enclosed uniformly can be a liquid or solid. 

Cylinder with Radius r and Height h

Types of Cylinders

The volume of Cylinders are classified into the following categories: 

1. Right Circular Cylinder: This type of cylinder has two bases as circles and each line that is part of the lateral curved surface is perpendicular to the base.
2. Oblique Cylinder: Oblique Cylinder has its sides leaning over the base at an angle that is not a right angle. 
3. Elliptic Cylinder: In an Elliptic Cylinder, the bases are not circular discs but are ellipses.
4. Right Circular Hollow Cylinder: Right Circular Hollow Cylinder has two right circular cylinders bound one inside the other. The volume of a cylinder is the exact capacity the cylinder can enclose or hold inside it. Thus, this is equivalent to the amount of space occupied by the cylindrical shape.

Surface Areas of Cylinders

Total surface area and curved surface area are two types of surface area of ​​a cylinder. Mathematically, the formula is given as: 

• Total Surface Area of a Cylinder = 2πr² + 2πr (r+h), and 
• Curved Surface Area of a Cylinder = 2πrh

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Volume of a Cylinder

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The volume of cylinder is measured by the number of cubic units of unit length that can fit inside the cylinder. This is equivalent to the three-dimensional space enclosed by the boundary of the cylinder. The volume of a cylinder is measured in cubic units like cm^​​3, m^​3, in^​3, etc. 

• The volume of a cylinder can be visualized by considering the cylinder as a collection of many congruent discs.
• These discs when stacked one above the other occupy the space inside the cylinder.
• Thus calculating the space occupied by each of these discs and adding them up, give the volume of a cylinder.
• The volume can thus be obtained by:

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Volume of a Cylinder Formula

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 The volume of a cylinder formula can be given by the product of the area of base and height. Mathematically, Volume of a cylinder (V) can be represented as

\(V=A\times H\)

Where

A= Area of the base in square units

H= Height of the cylinder measured in length units

The following are the volume formula for different types of cylinders

Volume of a Right Circular Cylinder

We know that the base of a right circular cylinder is a circle and the area of a circle of radius 'r' is For a right circular cylinder, the base is a circle with area as πr2. The volume of the right circular cylinder formula is:

\(V=πr^2\times h\)

Where,

r = base radius

h = Cylinder height

Π = constant = 22/7 or 3.14

The volume is dependent on the height and the square of the radius. 

Volume of an Oblique Cylinder

The volume of an oblique cylinder is the same as the right circular cylinder. The volume of an oblique cylinder formula with radius ‘r’ and height ‘h’ is given by:

\(V=πr^2\times h\)

Volume of an Elliptic Cylinder

An Elliptic cylinder had two radii as the bases are elliptical in shape. The area of the ellipse with radii a and b is πab. The volume of the elliptic cylinder formula is

 \(V=πab h\)

Where,

Radii of the ellipse of the cylinder - a and b

Height of the elliptical cylinder -h

Π = constant = 22/7 or 3.14

Volume of a Right Circular Hollow Cylinder

For a right circular cylinder that is consists of two right circular cylinders bounded one inside the other, the volume of a cylinder formula is obtained by subtracting the volume of the inside cylinder from that of the outside cylinder. Mathematically, 

\(V=π(R^2-r^2)\times h\)

R is the outer base radius of the cylinder

r is the inner base radius of the cylinder

h is the height of the cylinder

π is a constant= 22/7 = 3.14

Volume of Different Cylinder Types

Read More: NCERT Solutions Class 10 Surface Areas & Volumes

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How to Calculate Volume of a Cylinder?

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Following are quick steps to calculate the volume of cylinder:

1. Identify the radius and height of the given cylinder.
2. Ensure both quantities are in the same units, if not convert them into a common unit system.
3. Use the volume of a cylinder formula: \(V=πr^2\times h\). Substitute the values and determine the volume. 
4. The volume of a cylinder should be expressed in cubic units. 

Read More: Properties of Cylinders

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Conversions Units for Volume of a Cylinder

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Volume of a cylinder is expressed in cubic units like cm3, m3, dm3, etc. Now, thousand cubic centimeters is equal to one litre. So to convert from litre to cm3, 1000 is multiplied to the number. Similarly, to convert cm3 to litre, the number is divided by 1000. 

 For example,

1 Litre = 1000 cubic cm or cm^​3

1 cm^​3 = 1/1000 litres

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Solved Examples

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A few solved examples are illustrated below for better understanding of how to find volume of a cylinder.

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Things to Remember

• A cylinder is a three-dimensional shape with its bases circular, parallel, and congruent with one another.
• Cylinder can be visualized as a series of cubical units of unit length stacked together over one another. 
• The volume occupied by the cylinder is given by the product of the sum of the areas occupied by the cubic units and the total height of the cylinder.
• The volume of a cylinder formula is given by \(V=πr^2\times h\)
• The volume of an elliptical cylinder is given by : \(V=πab h\)
• The volume of a right circular hollow cylinder is given by: \(V=π(R^2-r^2)\times h\)

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