Volume of Cylinder: Definition, Properties, Formulas

The volume of a cylinder is the capacity of a cylinder that calculates the amount of material carried in it. The cylinder volume is measured using its radius and height. This article will discuss two types of cylinders: solid cylinders and hollow cylinders. Read below to know more about the formula, surface area and solved examples.

Volume of Cylinder: Introduction

A cylinder is a three-dimensional geometric object with one curved surface and two circular flat surfaces at the ends. A cylinder has three faces, one curved face and two flat faces, two edges (where two faces meet) and zero vertices (corners where two edges meet).

What is a Right Circular Cylinder?

A right circular cylinder is an object formed by rolling the rectangle on one of its sides as an axis.

If the axis (one of the sides of the rectangle) is perpendicular to the radius \((r),\) then the cylinder is called a right circular cylinder.

The base and top of the cylinder are circular in shape and they are parallel to each other, the distance between these circular faces of a cylinder is known as the height \(\left( h \right)\) of a cylinder.

Some Examples of Cylindrical Shape: Drinks can, battery, candles, water bottle, gas cylinder are few examples for cylindrical shape.

What is Volume?

Volume is the space occupied by the matter (solid, liquid, gas) inside the three-dimensional object or the volume of a three-dimensional object is generally defined as the capacity of the object, which can hold the matter.

How to Calculate Volume of Three-Dimensional Shape?

In general, the volume of three-dimensional shape is a product of its area of base and height.

What is the Volume of a Cylinder?

The volume of a cylinder is the measure of the amount of space occupied by matter inside a cylinder or the measure of the capacity of a cylinder.

Volume of a Cylinder Formula

The volume of a cylinder is equal to the product of the area of the circular base and the height of the cylinder.

Volume of a cylinder \( = \) Area of circle \( \times \) Height of a Cylinder
Area of circle, \(A = \pi {r^2}\)
Height of the right circular cylinder is \(h.\)
Volume of a cylinder \(= \pi {r^2} \times h = \pi {r^2}h\)

Volume of a Cylinder with Diameter

Volume of a cylinder with diameter \(= \pi {\left( {\frac{d}{2}} \right)^2}h = \frac{1}{4}\pi {d^2}h\)

Volume of a Cylinder Units

The units of volume of the cylinder are cubic millimetre \(\left( {{\rm{m}}{{\rm{m}}^3}} \right),\) cubic centimetre \(\left( {{\rm{c}}{{\rm{m}}^3}} \right),\) cubic meter \(\left( {{{\rm{m}}^3}} \right),\) etc.

Applications

The cylinder volume formula is useful in calculating the capacity or volume of cylindrical objects we use in our daily life. The volume of a cylinder formula helps in designing cylindrical objects as per the people need.

For example:

1. Water bottles
2. Oxygen cylinders in a hospital
3. Cylindrical lab flasks, etc.

The volume of any cylindrical container can be easily calculated using its radius and height.

Surface Area of a Cylinder

Curved surface area \(\left( {{\bf{CSA}}} \right)\) of a cylinder: The curved surface area of a cylinder is a measure of the only curved surface area that the cylinder occupies, leaving flat circular surfaces.

Curved surface area \(\left( {{\bf{CSA}}} \right)\) of a cylinder formula: Curved surface area \(\left( {{\rm{CSA}}} \right)\) of a cylinder \(= 2\pi rh\)

Total surface area \(\left( {{\bf{TSA}}} \right)\) of a cylinder: The total surface area of a cylinder is a measure of the total area that the surface of a cylinder occupies.

Total surface area \(\left( {{\bf{TSA}}} \right)\) of a cylinder formula: Total surface area \(\left( {{\rm{TSA}}} \right)\) of a cylinder \( = 2\pi rh + 2\pi {r^2} = 2\pi r\left( {h + r} \right).\)

What is a Hollow Cylinder?

A hollow cylinder is a cylinder that is empty from the inside and has differences in the outer and the inner surface radius of a cylinder.

\(‘r’\) is the outer radius of a hollow cylinder.
\(‘t’\) is the thickness of a hollow cylinder wall.
Thickness \(\left( t \right)\) of hollow cylinder wall \(=\) Outer surface radius of a hollow cylinder \( – \) Inner surface radius of a hollow cylinder.
\(‘h’\) is the height of a hollow cylinder.

Some examples of hollow cylinders: Pipes, straws, and paper rolls are a few examples of hollow cylinders.

Volume of a Hollow Cylinder

The volume of a hollow cylinder is equal to the product of the area of the circular ring (base) and the height of the hollow cylinder. The volume of a hollow cylinder is measured in cubic units​.

\({‘r_1’}\) is the radius of the inner surface of a hollow cylinder.
\({‘r_2’}\) is the radius of the outer surface of a hollow cylinder.
\(‘h’\) is the height of a hollow cylinder.

Volume of a Hollow Cylinder: Formula

The base of the hollow cylinder is a circular ring.

Area of a circular ring \(= \) Area of an outer circle \(-\) Area of an inner circle.
Area of a circular ring \(= \pi r_2^2 – \;\pi r_1^2 = \pi \left( {r_2^2 – r_1^2} \right)\)
Volume of a hollow cylinder \(= \) Area of a circular ring × Height of a hollow cylinder
Volume of a hollow cylinder \(= \;\pi \left( {r_2^2 – r_1^2} \right) \times h\)

Solved Examples

Question 1: The base radius of a cylindrical milk bottle is \(2\,{\rm{cm}}\) and height is \(5\,{\rm{cm}},\) calculate volume of milk bottle can hold?
Solution: Given:
Radius \(\left( r \right)\) of base \(= 2\,{\rm{cm}}\) and height \(\left( h \right) = 5\,{\rm{cm}}\)

We know that volume of a cylinder \(= \pi {r^2}h\)
\(= \;\frac{{22}}{7} \times 2 \times 2 \times 5\,{\rm{c}}{{\rm{m}}^3} = \;62.86\,{\rm{c}}{{\rm{m}}^3}\)
Hence, the volume of a cylindrical milk bottle is \(62.86\,{\rm{c}}{{\rm{m}}^3}.\)

Question 2: The volume of a cylindrical water tank is \(1100\,{{\rm{m}}^3},\) and the radius of the base of the cylindrical tank is \({\rm{5}}\,{\rm{m}}{\rm{.}}\) Calculate the height of the tank.
Solution: Given:
> Radius \(\left( r \right)\) of base \(= 5\;{\rm{m}}\) and volume of tank \(= 1100\,{\rm{c}}{{\rm{m}}^3}.\)
We know that volume of a cylinder \(= \pi {r^2}h\)
\(\Rightarrow 1100\,{\rm{c}}{{\rm{m}}^3} = \frac{{22}}{7} \times 5 \times 5 \times h\,{\rm{c}}{{\rm{m}}^2}\;\)
\(\Rightarrow h = 14\,{\rm{cm}}\)
Hence, the height of the tank is \(14\,{\rm{cm}}.\)

Question 3: The volume of a cylinder is \(220\,{{\rm{m}}^3},\) and the height of the cylinder is \(10{\mkern 1mu} \,{\rm{m}}.\) Find the radius of base of a cylinder.
Solution: Given:
Height \(\left( h \right)\) of the cylinder \({\rm{ = 10}}\,{\rm{m}}\) and volume of a cylinder \(= 220\,{{\rm{m}}^3}\)
We know that volume of a cylinder \(\; = \pi {r^2}h\)
\(\Rightarrow 220\,{{\rm{m}}^3} = \;\frac{{22}}{7}\; \times {r^2} \times 10\,{\rm{m}}\)
\(\Rightarrow {r^2} = 7\,{{\rm{m}}^2}\)
\(\Rightarrow r = \sqrt 7 \,{\rm{m}}\)
Hence, the radius of a cylinder is \(\sqrt 7 \,{\rm{m}}.\)

Question 4: Find the volume of a cylindrical coffee mug, whose base radius is \({\rm{4}}\,{\rm{cm}}\) and height is \({\rm{7}}\,{\rm{cm}}{\rm{.}}\)
Solution: Given:
Base radius of a cylindrical coffee mug \(= 4\;{\rm{cm}}\)
Height of a cylindrical coffee mug \(= 7{\rm{\;cm}}\)
We know that volume of a cylinder \(= \pi {r^2}h\)
Now, volume of a cylindrical coffee mug \(= \frac{{22}}{7}\; \times 4\,{\rm{cm}} \times 4\,{\rm{cm}} \times 7\,{\rm{cm}} = 352\,{\rm{c}}{{\rm{m}}^3}\)
Hence, the volume of a cylindrical coffee mug is \(352\;{\rm{c}}{{\rm{m}}^3}.\)

Question 5: A solid cylinder of base radius is \(10\,{\rm{cm}},\) and the height is \({\rm{7}}\,{\rm{cm,}}\) is melted and re-casted into small cubes of edge \({\rm{2}}\,{\rm{cm}}{\rm{.}}\) Find the number of cubes can be formed?
Solution: Given:
The base radius of a solid cylinder \( = 10\;{\rm{cm}}\)
The height of solid cylinder \( = 7\;{\rm{cm}}.\)
Edge \(\left( a \right)\) of the cube \( = 2{\rm{\;cm}}\)
We know that volume of a cylinder \( = \pi {r^2}h\)
Now, volume of a solid cylinder \( = \frac{{22}}{7}\; \times 10 \times 10 \times 7\;{\rm{c}}{{\rm{m}}^3} = 2200\;{\rm{c}}{{\rm{m}}^3}\)
We know that volume of a cube \( = {a^3}\)
Now, volume of small cube \( = {2^3} = 8\;{\rm{c}}{{\rm{m}}^3}\)
Therefore, number of small cubes formed \( = \frac{{2200{\rm{\;c}}{{\rm{m}}^3}}}{{8{\rm{\;c}}{{\rm{m}}^3}}} = 275\)
Hence, the number of small cubes formed is \(275.\)

Question 6: Find the volume of a cylindrical metal pipe, whose length is \({\rm{1}}\,{\rm{m}}\) and outer radius is \(20\;{\rm{cm}},\) and the thickness of the metal pipe is \(5\;{\rm{cm}}.\)
Solution: Given:
Length \(\left( h \right)\) of the metal pipe \( = 1\;{\rm{m}} = 100\;{\rm{cm}}\)
Outer radius of the pipe \( = 20\;{\rm{cm}}\)
Inner radius of the pipe \(=\) Outer radius of the pipe \(-\) Thickness of the metal pipe
Inner radius of the pipe \( = 20 – 5 = 15\;{\rm{cm}}\)
We know that volume of a hollow cylinder \( = \pi \left( {{r_2}^2 – {r_1}^2} \right) \times h\)
Now, the volume of a metal pipe \(\; = \frac{{22}}{7} \times \left( {{{20}^2} – {{15}^2}} \right) \times 100{\rm{\;c}}{{\rm{m}}^3}\)
\( = \;\frac{{22}}{7}\; \times 175 \times 100\;{\rm{\;c}}{{\rm{m}}^3} = 55000\;{\rm{\;c}}{{\rm{m}}^3}\)
Hence, the volume of a metal pipe is \(55000\;{\rm{\;c}}{{\rm{m}}^3}.\)

Question 7: A metal pipe of inner radius \({\rm{4}}\,{\rm{cm,}}\) outer radius \({\rm{6}}\,{\rm{cm}}\) and length of a pipe is \({\rm{7}}\,{\rm{cm}}\) is melted and re-casted into a solid cylinder with radius \({\rm{2}}\,{\rm{cm,}}\) find the height of solid cylinder.
Solution: Given:
Inner radius \(\left( {{r_1}} \right)\) of metal pipe \( = 4{\rm{\;cm}}\)
Outer radius \(\left( {{r_2}} \right)\) of metal pipe \( = 6\;{\rm{cm}}\)
Length \(\left( h \right)\) of a pipe \( = \;7\;{\rm{cm}}\)
We know that volume of a hollow cylinder \( = \pi \left( {{r_2}^2 – {r_1}^2} \right) \times h\)
Volume of metal pipe \( = \frac{{22}}{7}\; \times \left( {{6^2} – {4^2}} \right) \times 7{\rm{\;c}}{{\rm{m}}^3}\)
Volume of metal pipe \( = 440\;{\rm{c}}{{\rm{m}}^3}\)
Now, we need to find the height of the solid cylinder.
Given, the radius of the solid cylinder is \(2\;{\rm{cm}}.\)
We know that volume of a cylinder \( = \pi {r^2}h\)
Here, Volume of a metal pipe \(=\) Volume of solid cylinder
\( \Rightarrow 440\;{\rm{c}}{{\rm{m}}^3} = \frac{{22}}{7}\; \times 2 \times 2 \times h\;{\rm{c}}{{\rm{m}}^2}\)
\( \Rightarrow h = 35\;{\rm{cm}}\)
Hence, the height of the solid cylinder is \(35\;{\rm{cm}}.\)

Question 8: The volume of a hollow cylinder is \(660\,{\rm{c}}{{\rm{m}}^3}.\) If the outer radius is \({\rm{12}}\,{\rm{cm}}\) and inner radius is \({\rm{10}}\,{\rm{cm}}{\rm{.}}\) Find the height of a hollow cylinder.
Solution: Given:
The volume of a hollow cylinder \( = 660\;{\rm{c}}{{\rm{m}}^3}\)
The outer radius of a hollow cylinder \( = 12\;{\rm{cm}}\)
The inner radius of a hollow cylinder \( = 10\;{\rm{cm}}\)
We know that volume of a hollow cylinder \( = \pi \left( {{r_2}^2 – {r_1}^2} \right) \times h\)
Now, \(660\;{\rm{c}}{{\rm{m}}^3} = \;\frac{{22}}{7}\; \times \;\left( {{{12}^2} – {{10}^2}} \right)\; \times h\;{\rm{c}}{{\rm{m}}^2}\)
\( \Rightarrow 660\;{\rm{c}}{{\rm{m}}^3} = \frac{{22}}{7}\; \times \left( {144 – 100} \right) \times h\;{\rm{c}}{{\rm{m}}^2}\)
\( \Rightarrow 660\;{\rm{c}}{{\rm{m}}^3} = \;\frac{{22}}{7}\; \times \;44\; \times \;h{\rm{\;c}}{{\rm{m}}^2}\)
\( \Rightarrow h = \frac{{660 \times 7}}{{22 \times 44}}\;{\rm{cm}}\)
\( \Rightarrow h = 4.8\;{\rm{cm}}\)
Hence, the height of a hollow cylinder is \(4.8{\rm{\;cm}}.\)

Question 9: Volume of a hollow cylinder is \(300\;{\rm{c}}{{\rm{m}}^3},\) if its outer radius is \({\rm{4}}\,{\rm{cm}}\) and its height is \({\rm{7}}\,{\rm{cm,}}\) the find its inner radius.
Solution: Given:
Volume of a hollow cylinder \( = 300\;{\rm{c}}{{\rm{m}}^3}\)
Outer radius of a hollow cylinder \( = 4\;{\rm{cm}}\)
The height of a hollow cylinder \( = 7\;{\rm{cm}}\)
We know that volume of a hollow cylinder \( = \pi \left( {{r_2}^2 – {r_1}^2} \right) \times h\)
Now, \(300\;{\rm{c}}{{\rm{m}}^3} = \frac{{22}}{7}\; \times \left( {{4^2} – {r_1}^2} \right) \times 7\;{\rm{c}}{{\rm{m}}^3}\)
\(\Rightarrow 300\;{\rm{c}}{{\rm{m}}^3} = 352\; – – 22\;r_1^2\)
\(\Rightarrow 22\;{r_1}^2 = 352{\rm{ – }}300\)
\( \Rightarrow {r_1} = \sqrt {\frac{{52}}{{22}}} = \sqrt {2.4} \;{\rm{cm}}\)
Hence, the inner radius of the hollow cylinder is \(\sqrt {2.4} \;{\rm{cm}}.\)

Question 10: Find the volume of the wood used in the pencil, if its outer radius is \({\rm{4}}\,{\rm{mm,}}\) its inner radius is \({\rm{3}}\,{\rm{mm}}\) and its height is \({\rm{50}}\,{\rm{mm}}{\rm{.}}\)
Solution: Given:
Outer radius of a pencil \( = 4\;{\rm{mm}}\)
Inner radius of a pencil \( = 3\;{\rm{mm}}\)
Height of a pencil \( = 50\;{\rm{mm}}\)
We know that volume of a hollow cylinder \( = \pi \left( {{r_2}^2 – {r_1}^2} \right) \times h\)
Now, the volume of the wood used in pencil \( = \frac{{22}}{7}\; \times \left( {{4^2}–{3^2}} \right) \times 50\;{\rm{m}}{{\rm{m}}^3}\)
\( = \;\frac{{22}}{7}\; \times 7\; \times 50\;\;{\rm{m}}{{\rm{m}}^3} = 1100\;{\rm{m}}{{\rm{m}}^3}\)
Hence, the volume of the wood used in pencil is \(1100\;{\rm{m}}{{\rm{m}}^3}.\)

Few conversions in volume of a cylinder:

\(1\,{\rm{litre}} = 0.001\) cubic meter \(\left( {{{\rm{m}}^3}} \right)\)

Example:
\({\rm{55}}\,{\rm{litres}}\) of water is equal to \(0.055\;{{\rm{m}}^3}.\)
\(1\,{\rm{litre}}\,{\rm{ = 1000}}\) cubic centimetre \({{\rm{(cm)}}^{\rm{3}}}\)
\({\rm{1}}\,{\rm{litre = 1000}}\) cubic centimetre \({{\rm{(cm)}}^{\rm{3}}}\)
Example:
\({\rm{32}}\,{\rm{litres}}\) of milk is equal to \(32000\;{\rm{c}}{{\rm{m}}^3}.\)

Summary

In this article, we learnt about the cylinder, right-circular cylinder, volume, formula to calculate the volume of a cylinder, some applications of the volume of a cylinder formula, hollow cylinder, volume of a hollow cylinder, formula to calculate the volume of a hollow cylinder.

The learning outcome from the volume of a cylinder will help in understanding all the concepts related to cylinders, how the cylindrical objects are designed, etc.

FAQs

Q.1. What is the unit for volume of hollow cylinders?
Ans: Units for the volume of the hollow cylinder are cubic meter \(\left( {{{\rm{m}}^3}} \right)\) or cubic centimetre \(\left( {{\rm{c}}{{\rm{m}}^3}} \right)\) etc.

Q.2. What is the volume of a cylinder?
Ans: Volume of a cylinder is the space occupied by a matter inside the cylinder.

Q.3. What is the formula to find the volume of a cylinder?
Ans: Formula to find the volume of a cylinder is \(\pi {r^2}h\)

Q.4. How to find the volume of a cylinder, when area of base and height of a cylinder is given?
Ans: Volume of a cylinder is a product of its base and height. So, when the area of base and height of a cylinder is given, find the product of them to get the volume of a cylinder.

Q.5. How to find the volume of a cylinder, when the diameter of base and height of the cylinder is given?
Ans: Diameter is twice the radius, find the radius of base and put the values of radius of a base and height of a cylinder formula and calculate the volume.

Q.6. How to find the radius of the cylinder, when volume of a cylinder and height of a cylinder is given?
Ans: Rearrange the volume of a cylinder formula to find the radius of a cylinder.
That is, \(r = \sqrt {\frac{{{\rm{volume\;of\;a\;cylinder}}}}{{\pi h}}} \)

Q.7. How to find the height of the cylinder, when the volume of a cylinder and radius of a cylinder is given?
Ans: Rearrange the volume of a cylinder formula to find radius of a cylinder. That is, \(h = \frac{{{\rm{volume \;of \;a \;cylinder }}}}{{\pi {r^2}}}\)

Q.8. Why is the volume of a cylinder formula is \(\pi {r^2}h\)?
Ans: The volume of a cylinder is equal to the product of the area of the circular base and the height of the cylinder.
That is, \({\rm{volume \;of \;a \;cylinder}} = {\rm{Area \;of \;circle}} \times {\rm{Height}}\)
Volume of a cylinder \( = \pi {r^2} \times h = \;\pi {r^2}h\)

Q.9. What is the formula to find the volume of a hollow cylinder?
Ans: Volume of a hollow cylinder \( = \pi \left( {{r_2}^2 – {r_1}^2} \right) \times h\) Where,
\({‘r_1’}\) is the radius of the inner surface of a hollow cylinder.
\({‘r_2’}\) is the radius of the outer surface of a hollow cylinder.
\(‘h’\) is the height of a hollow cylinder.

Q.10. How to find the thickness of metal used in hollow cylinders?
Ans: Thickness of metal in hollow cylinders is the difference of outer surface radius \(\left( {{r_2}} \right)\) and inner surface radius \(({r_1})\) of a hollow cylinder.
That is, the thickness of a metal \(\; = {r_2}\;–{r_1}.\)

Q.11. How to find the volume of metal if the inner radius, the thickness of metal and height of a hollow cylinder is given?
Ans: Thickness of metal in hollow cylinders is the difference between the outer surface radius and inner surface radius of a hollow cylinder.
That is, thickness of a metal \(\; = {r_2}\;–{r_1},\) using this find the outer radius of a hollow cylinder and put outer radius, inner radius, and height of hollow cylinder in the volume of a hollow cylinder formula calculate the volume of metal.

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