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Volume of a Cylinder(Formulas, Derivation & Examples)

volume of a cylinder

A cylinder is a 3D shape with a curved surface and two flat circular surfaces. One of the most common real-world examples of a cylinder is an LPG cylinder.

The space occupied by a cylinder is called the volume of a cylinder. It’s also referred to as the capacity of a cylinder and it depends on the radius and the height of the cylinder. Volume of a cylinder is measured in terms of cubic units, such as $cm^{3}$, $m^{3}$, $mm^{3}$, $in^{3}$, $ft^{3}$, etc.

Let’s learn how to find the volume of a cylinder and its formula.

Cylinder – A 3D Solid Shape

A cylinder is one of the basic 3D shapes, in geometry, which has two parallel circular bases at some distance. The two circular bases are joined by a curved surface, at a fixed distance from the centre. The line segment joining the centre of two circular bases is the axis of the cylinder. The distance between the two circular bases is called the height of the cylinder. LPG gas cylinder is one of the real-life examples of cylinders. 

Since the cylinder is a three-dimensional shape, therefore it has two major properties, i.e., surface area and volume. The space occupied by a cylinder in three dimensions is called its volume and it is equal to $\pi r^{2} h$. 

Some of the important properties of the cylinder are as follows:

  • The bases of the cylinder are always congruent and parallel to each other.
  • If the axis of the cylinder is at a right angle to the base and the bases are exactly over each other, then it is called a “Right Cylinder”.
  • If one of the bases of the cylinder is displayed sideways, and the axis does not produce the right angle to the bases, then it is called an “Oblique Cylinder”.
  • If the bases are circular, then it is called a right circular cylinder.
  • The best alternative to the circular base of a cylinder is an ellipse. If the base of the cylinder is elliptical in shape, then it is called an “Elliptical Cylinder”.
  • If the locus of a line moves parallel and fixed distance from the axis, a circular cylinder is produced.
  • A cylinder is similar to a prism since it has the same cross-section everywhere.
volume of a cylinder

For ease of understanding, the right cylinder or right circular cylinder is considered for studying the different properties of the cylinder.

Right Circular Cylinder

A cylinder whose bases are circular in shape and parallel to each other is called the right circular cylinder. It is a three-dimensional shape. The axis of the cylinder joins the center of the two bases of the cylinder. This is the most common type of cylinder used in day-to-day life. It is different from the oblique cylinder which does not have parallel bases and resembles a tilted structure.

volume of a cylinder

Parts of Right Circular Cylinder

The three parts of the right circular cylinder are:

  • Top Circular Base
  • Curved Lateral Face
  • Bottom Circular Base

Properties of Right Circular Cylinder

These are the properties of a right circular cylinder.

  • The line joining the centers of the circle is called the axis.
  • When we revolve a rectangle about one side as the axis of revolution, a right cylinder is formed.
  • The section obtained on cutting a right circular cylinder by a plane contains two elements and the parallel to the axis of the cylinder is the rectangle.
  • If a plane cuts the right cylinder horizontally parallel to the bases, then it’s a circle.

What is the Volume of a Cylinder?

The volume of a cylinder is the number of cubic units, occupied by the cylinder completely. The unit of volume of a cylinder is cubic units such as $cm^{3}$, $m^{3}$, $mm^{3}$, $in^{3}$, $ft^{3}$.

volume of a cylinder

If the measure of radius and height of a cylinder are $r$ and $h$ unit, then the formula for calculating its volume is $\pi r^{2} h$.

Derivation of Volume of a Cylinder Formula

A right circular cylinder can be considered as a $3D$ shape formed by keeping a number of circles one over the other.

volume of a cylinder

Let’s consider a cylinder of base radius $r$ and height $h$ formed by a heap(or stack) of height $h$ formed by circles of radius $r$.

The volume of this cylinder will be the sum of the areas of all these circles.

Area of a circle of radius $r$ is $\pi r^{2}$.

Therefore, the sum of the area of all circles will be $\pi r^{2} + \pi r^{2} + \pi r^{2} + … + \left( h \text{ times} \right) = \pi r^{2} \times h = \pi r^{2}h$.

The formula for the volume of a cylinder is $\pi r^{2}h$.

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Volume of Material Used in a Cylinder

Consider a cylinder made from a metallic sheet of a certain thickness such that the internal radius of the cylinder is $r$ and the external radius is $R$ and the height $h$ units.

Note: The thickness of the metallic sheet is $R – r$.

volume of a cylinder

The volume of outer cylinder with radius $R$ and height $h$ is $V_{O} = \pi R^{2} h$.

And the volume of inner cylinder with radius $r$ and height $h$ is $V_{I} = \pi r^{2} h$.

The amount of material used in constructing the cylinder is $V_{O} – V_{I} = \pi R^{2} h – \pi r^{2} h =  \pi \left(R^{2} – r^{2} \right)h$.

The formula for the amount of material used in making a cylinder is $ \pi \left(R^{2} – r^{2} \right)h$

where $R$ is the outer radius

$r$ is the inner radius

$h$ is the height

Thickness of material = $R – r$

Examples

Ex 1: Calculate the volume of a cylinder where the area of the base is $15 cm^{2}$ and the height is $8 cm$.

Area of the base of a cylinder $A = 15 cm^{2}$

Height of a cylinder $h = 8 cm$

Volume $V = A \times h = 15 \times 8 = 120 cm^{3}$.

Note: Volume $V = Ah$, where $A$ is the cross-section area and $h$ is the height of a 3D object.

Ex 2: Calculate the volume of a cylinder where the radius of the base is $10.5 cm$ and the height is $5 cm$.

Radius of the base of a cylinder $r = 10.5 cm$

Height of a cylinder $h = 15 cm$

Volume of a cylinder = $\pi r^{2} h = \frac {22}{7} \times 10.5^{2} \times 5 =1732.5 c.c.$.

Note: $c.c.$ means cubic centimetre $cm^{3}$.

Ex 3: Given the internal radius of the pipe is $2 cm$, the external radius is $2.4 cm$ and the length of the pipe is $10 cm$. Find the volume of the metal used.

The internal radius of the cylindrical pipe $r = 2 cm$

The external radius of the cylindrical pipe $R = 2.4 cm$

Length (Height) of the cylindrical pipe $h = 10 cm$

The volume of the metal used =$\pi \left(R^{2} – r^{2} \right)h = \frac {22}{7} \times \left(2.4^{2} – 2^{2} \right) \times 10 = \frac {22}{7} \times \left(5.76 – 4 \right) \times 10 = 55.31 cm^{3}$. (Rounded off to $2$ decimal places)

Ex 4: Which of the following can hold more fluid? A cubical container of length of edge $10 cm$ or a cylindrical container of diameter $10 cm$ and height $10 cm$.

Dimensions of a cube: Edge length $a = 10 cm$.

Volume of a cube = $a^{3} = 10^{3} = 1,000 c.c$

Dimensions of a cylinder: Height $h = 10 cm$ and diameter $d = 10 cm$ or radius $r = \frac {10}{2} = 5 cm$

Volume of a cylinder = $\pi r^{2} h = \frac {22}{7} \times 5^{2} \times 10 = 785.71 cm^{3}$ (Rounded off to $2$ decimal places).

Therefore, a cubical container of edge length $10 cm$ can hold more fluid as compared to a cylindrical container of diameter $10 cm$ and height of $10 cm$.

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Conclusion

The volume of a cylinder is the number of cubic units, occupied by the cylinder completely and is calculated by using its radius and height. If $r$ is the radius and $h$ is the height of a cylinder, then the formula for computing its volume is $\pi r^{2} h$.

Practice Problems

  1. A cylinder has a radius of $r=3.5 cm$ and a height of $h = 30cm$.  What is its volume?
  2. What is the volume of a cylinder that has a base with a radius of $7 in$ and a height of $50 in$?
  3. What is the volume of a cylinder with a radius of $10.5 mm$ and a length that is three times as long as its diameter?
  4. Water glass has the shape of a right cylinder. The glass has an interior radius of $7 cm$, and a height of $10 cm$. The glass is $75 \%$ full. What is the volume of the water in the glass?
  5. A circle has a circumference of $4 \pi$ and it is used as the base of a cylinder. The cylinder has a surface area of $16 \pi$. Find the volume of the cylinder.

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FAQs

What is the volume of a cylinder?

The volume of a cylinder is the amount of space in it. It can be obtained by multiplying its base area by its height. The volume of a cylinder of base radius $r$ and height $h$’ is $\pi r^{2}h$.

How do you find the volume of a cylinder?

The formula used to find the volume of a cylinder of base radius $r$ and height $h$ is $\pi r^{2} h$.

How the amount of material used in a cylinder is calculated?

The formula for the amount of material used in a cylinder is $ \pi \left(R^{2} – r^{2} \right)h$
where $R$ is the outer radius
$r$ is the inner radius
$h$ is the height
Thickness of material = $R – r$

What is the volume of a cylinder using diameter?

Let us consider a cylinder of radius $r$, diameter $d$, and height $h$. The volume of a cylinder of base radius $r$ and height $h$ is $V = \pi r^{2}h$. 

We also know that $r = \frac {d}{2}$. 

By substituting this in the above formula we get $V = \pi \frac {d^{2}h}{4}$.

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