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# Cylinder

## Introduction

In our daily lives, we come across several solids like jars, gas cylinders etc. Have you noticed that such solids have curved surfaces with congruent circular ends? Such solids are known as cylinders. Objects such as a circular pillar, a circular pipe, a test tube, a circular storage tank, a measuring jar, a gas cylinder, a circular powder tin etc. are all shapes of a cylinder. Let us learn more about cylinders.

## Definition

A Cylinder is one of the basic 3d shapes, in geometry, which has two parallel circular bases at a distance. In other words, a cylinder is a 3D solid shape that consists of two identical and parallel bases linked by a curved surface. Let us recall that 3 Dimensional shapes or 3D shapes are the shapes that have all three dimensions, i.e. length, breadth and height.  A right circular cylinder, being a 3 shape has two plane ends. Each plane end is circular in shape, and the two plane ends are parallel; i.e. they lie in parallel planes. Each of the plane ends is called a base of the cylinder. Each of the plane end is called the base of the cylinder. Below we have a general diagram of a Cylinder –

Let us now learn about some basic terms that are integral to the understanding of a cylinder –

Base – Each of the circular ends on which the cylinder rests is called its base.

Axis – The line segment joining the centres of two circular bases is called the axis of the cylinder.

Height – The length of the axis of the cylinder is called the height of the cylinder.

Lateral Surface – the curved surface joining the two bases of a right circular cylinder is called its lateral surface.

Faces – A cylinder has one curved surface and two flat faces. In other words, a cylinder has two plane faces, namely the top and the base and one curved face.

Edges – A cylinder has two curved edges.

Vertx – A cylinder has no corner or vertex.

## Types of cylinder

There are four different types of cylinders based on the axis and the bases. These are –

1. Right circular cylinder
2. Oblique cylinder
3. Elliptic Cylinder
4. Right Circular Hollow Cylinder

Let us learn about them one by one.

### Right Circular Cylinder

A cylinder where the axis is perpendicular to the bases is called a right circular cylinder.

### Oblique Cylinder

A cylinder where the axis is not perpendicular to the bases is called an oblique cylinder.

### Elliptic Cylinder

A cylinder whose base is in the form of an ellipse is called an elliptic cylinder.

### Right Circular Hollow Cylinder

Right circular hollow cylinder consists of two right circular cylinders that are bounded one inside the other. This cylinder has a common point of the axis and is perpendicular to the central base. Being hollow means that it is hollow from inside it is different from the right circular cylinder.

## Surface Area of a Right Circular Cylinder

Consider a right circular cylinder of radius r and height h. Area of the lateral surface of the cylinder is given by –

Area of the lateral surface of the cylinder

= Area of the rectangular strip of paper

= Area of a rectangle of length 2 π r and breadth h

= 2 π r x h square units

= 2 π r h square units

Thus, for a cylinder of radius r and height h we have,

Lateral ( curved ) surface area  = 2 π r h square units

Each base surface area = π r 2 square units

Total Surface Area = ( 2 π r h + 2 π r 2 ) square units = 2 π r ( h + r )  square units.

Let us understand it through an example.

Example

Find the curved surface area and total surface area of a right circular cylinder whose height is 15 cm and the radius of the base is 7 cm. ( Take π = 22/7 )

Solution

We have been given that

Radius of the base of the cylinder = r = 7 cm

Height of the cylinder = h = 15 cm

Now, we know that

Lateral ( curved ) surface area  of a cylinder = 2 π r h square units ………. ( 1 )

Substituting the values of π, r and h in the above equation, we get

Lateral ( curved ) surface area  of a cylinder = ( 2 x $\frac{22}{7}$ x 7 x 15 ) sq. cm

⇒ Lateral ( curved ) surface area  of a cylinder = ( 2 x 22 x 7 x 15 ) sq. cm

Lateral ( curved ) surface area  of a cylinder = 660 sq. cm

Next, we need to find the total surface area of the given cylinder. We know that

Total Surface Area of a cylinder = 2 π r ( h + r )  square units.

Substituting the values of π, r and h in the above equation, we get

Total Surface Area of a cylinder = 2 x $\frac{22}{7}$ x 7 x ( 15 + 7  ) sq. cm

⇒ Total Surface Area of a cylinder = 2 x 22x  15 sq. cm

Total Surface Area of a cylinder = 968 sq. cm.

## Surface Area of a Hollow Cylinder

Solids such as iron pipes, rubber tubes etc. are hollow cylinders. Thus we can say that a solid bounded by two coaxial cylinders of the same height and different radii is called a hollow cylinder. What would be the surface area of a hollow cylinder? Let us find out.

Let R and r be the external and internal radii of a hollow cylinder respectively and h be the height of the hollow cylinder.

Surface area of each base of the hollow cylinder =  π ( R 2 – r 2 ) sq. units

Curved ( lateral ) Surface area of a hollow cylinder = ( External surface area ) + ( internal surface area )

⇒  Curved ( lateral ) Surface area of a hollow cylinder = 2 π R h  + 2 π r h

⇒  Curved ( lateral ) Surface area of a hollow cylinder = 2 π h ( R + r ) sq. units

Similarly,

Total surface area of a hollow cylinder = 2 π R h + 2 π r h + 2 π ( R 2 – r 2 )

⇒ Total surface area of a hollow cylinder = 2 π h ( R + r ) + 2 π ( R + r ) ( R – r )

Total surface area of a hollow cylinder = 2 π ( R + r ) ( h + R – r ) sq. units

Let us understand it through an example.

Example

The lateral surface area of a hollow cylinder is 424 sq. cm. It is cut along its height and formed a rectangular sheet of width 32 cm. Find the perimeter of the rectangular sheet.

Solution

We have been given that the lateral surface area of a hollow cylinder is 424 sq. cm. It is cut along its height and formed a rectangular sheet of width 32 cm. we are required to find the perimeter of the rectangular sheet. Let us first summarise the information given to us.

It is important to understand that the height of the cylinder from the sheet would be equal to the width of the rectangular sheet. Therefore, we have,

Height of the cylinder = h = Width of the sheet = 32 cm

Now, let r be the radius of the base of the hollow cylinder.

We have been given that the lateral surface area of a hollow cylinder = 424 sq. cm …… ( 1 )

Also, we have learnt that

Lateral ( curved ) surface area  = 2 π r h square units

Now, h = 32 cm, π = $\frac{22}{7}$

Substituting the values of h and π in the above equation, we have,

Lateral ( curved ) surface area  = 2 x $\frac{22}{7}$ x r 32   ……… ( 2 )

From ( 1 ) and ( 2 ), we have,

2 x $\frac{22}{7}$ x r 32  = = 424

⇒ r = $\frac{424 x 7}{32 x 2 x 22}$

⇒ r = = 21 cm

Now, that we have obtained the value of the radius of the base, let us find the perimeter of the sheet.

We know that the perimeter of a circular sheet is given by 2 π r. Therefore, we have,

Perimeter of the sheet = 2 π r = 2 x $\frac{22}{7}$ x 21

⇒ Perimeter of the sheet = 2 x 22 x 3 = 132 cm

Hence, the perimeter of the sheet will be 132 cm

## Volume of a Cylinder

Let us take a right circular cylinder of radius r and height h. we know that the volume of a right circular cylinder is equal to the measure of the space occupied by the cylinder.

Therefore, we have,

Volume of a right circular cylinder = measure of the space occupied by the cylinder

⇒ Volume of a right circular cylinder = The area of each circular sheet x Height

Volume of a right circular cylinder = π r 2 h

Let us understand it through an example.

Example

The area of the base of a right circular cylinder is  154 sq. cm and its height is 15 cm. Find the volume of the cylinder.

Solution

We have been given that the area of the base of a right circular cylinder is  154 sq. cm and its height is 15 cm. We need to find the volume of the cylinder. Let us first summarise the information given to us. We have,

Area of the base of a right circular cylinder = 154 sq. cm

Height of the right circular cylinder = 15 cm

Now, we know that

Volume of a right circular cylinder = ( Area of the base ) x Height …… ( 1 )

Substituting the values available to us in the above equation we get,

Volume of a right circular cylinder = 154 x 15 cm 3

⇒ Volume of a right circular cylinder = 2310 cm 3

Hence, the volume of the cylinder having height 15 cm and area of its base as 154 sq. cm will be given by = 2310 cm 3

## Volume of a Hollow Cylinder

Recall that solids such as iron pipes, rubber tubes etc. are hollow cylinders. Thus we can say that a solid bounded by two coaxial cylinders of the same height and different radii is called a hollow cylinder. What would be the volume of a hollow cylinder? Let us find out.

Let R and r be the external and internal radii of a hollow cylinder respectively and h be the height of the hollow cylinder.

Volume of the hollow cylinder will be the difference between the exterior and the interior volumes. This means that –

Volume of hollow cylinder = ( Volume of exterior portion ) – ( Volume of interior portion )

Now, we know that –

Volume of exterior portion = π R 2 h and

Volume of interior portion = π r 2 h

Therefore,

Volume of hollow cylinder =  π R 2 h – π r 2 h

Volume of hollow cylinder =  π h (R 2 – r 2 )

Let us understand it through an example.

Example

The thickness of a metallic tube is 1 cm and the inner diameter of the tube is 12 cm. Find the weight of 1 m long tube, if the density of the metal be 7.8 g / cm ³.

Solution

We have been given that the thickness of a metallic tube is 1 cm and the inner diameter of the tube is 12 cm. also, the density of the metal is 7.8 g / cm ³. We need to find the weight of 1 m long tube. Let us first summarise the information given to us. We have,

Inner diameter of the tube = 12 cm

This means that the inner radius of the tube = r = 6 cm ……… ( 1 )

Also,

Thickness of the tube = 1 cm

Therefore, outer radius of the tube = R = ( 6 + 1 ) cm = 7 cm ………. ( 2 )

Length ( Height ) of the tube = h = 1 m = 100 cm …… ( 3 )

Now, we have learnt that the Volume of hollow cylinder =  π h (R 2 – r 2 ) …… ( 4 )

Substituting the values of r, R and h that we have obtained din the equations ( 1 ) , ( 2 )  and  ( 3 ) in the above equation, we have,

Volume of metal in the tube =  π h (R 2 – r 2 )

⇒ Volume of metal in the tube =  $\frac{22}{7}$ x 100 ( 7 2 – 6 2 ) cm 3

⇒ Volume of metal in the tube =  $\frac{22}{7}$ x 100 x 13 cm 3 ………. ( 5 )

Also, we have been given that density of the metal is 7.8 g / cm ³

Therefore,

Weight of the tube = Volume x Density

⇒ Weight of the tube =  $\frac{22}{7}$ x 100 x 13 x 7.8

⇒ Weight of the tube =  31868.57 g = $\frac{31868.57}{1000}$ kg = 31.86857 kg = 31.869 kg

Hence, the weight of the tube = 31.869 kg

## Key Facts and Summary

1. A cylinder is a 3D solid shape that consists of two identical and parallel bases linked by a curved surface.
2. Each of the circular ends on which the cylinder rests is called its base.
3. The line segment joining the centres of two circular bases is called the axis of the cylinder.
4. The radius of the circular bases is called the radius of the cylinder.
5. The length of the axis of the cylinder is called the height of the cylinder.
6. The curved surface joining the two bases of a right circular cylinder is called its lateral surface.
7. A cylinder has one curved surface and two flat faces.
8. A cylinder has two curved edges and has no corner or vertex
9. A cylinder where the axis is perpendicular to the bases is called a right circular cylinder.
10. A cylinder where the axis is not perpendicular to the bases is called an oblique circular cylinder.
11. A cylinder whose base is in the form of an ellipse is called an elliptic cylinder.
12. Right circular hollow cylinder consists of two right circular cylinders that are bounded one inside the other.
13. For a cylinder of radius r and height h we have, Lateral ( curved ) surface area  = 2 π r h square units, Each base surface area = π r 2 square units and Total Surface Area = 2 π r ( h + r )  square units.
14. A solid shape bounded by two coaxial cylinders of the same height and different radii is called a hollow cylinder.
15. Surface area of each base of the hollow cylinder =  π ( R 2 – r 2 ) sq. units
16. Curved ( lateral ) Surface area of a hollow cylinder = 2 π h ( R + r ) sq. units
17. Total surface area of a hollow cylinder = 2 π ( R + r ) ( h + R – r ) sq. units
18. Volume of a right circular cylinder = π r 2 h
19. Volume of hollow cylinder =  π h (R 2 – r 2 )