MENSURATION

Acknowledgement

We would like to express our special thanks to our Math's teacher who gave us the golden opportunity to do this wonderful presentation on the topic ‘Mensuration’. This helped us in doing a lot of research work and we came to know about many things, I am really grateful to the teacher for her constant guidance and support.

Contents

Introduction

Quote—Unquote

Important Terms

Figures

Measuring Plain Figures

Measuring Solid Figures

Review Of Formulae

Group Members

INTRODUCTION

• Mensuration is the branch of geometry which deals with the measurement of area, length or volume. It is also the act or process of measuring.

• The Mensuration took its birth in Egypt. Then it was applied and expanded by great people like Pythagoras, Euclid, Archimedes, Ptolemy etc and further developed by Halley, Bernouillies, Euler, Newton etc.

Quote--Unquote

Listen to some famous quotes from famous people about geometry.

I think the universe is pure geometry- basically, a beautiful shape twisting around space-time.

There is geometry in the humming of strings, there is music in the spacing of spheres.

Important Terms

• Solid: A body or geometric figure having three dimensions.

Important Terms

• Surface Area: The total area of the surface of the three dimensional figure.

Important Terms

• Perimeter: The continuous line forming the boundary of a geometrical figure.

Important Terms

• Volume: The total space occupied by an object or the space inside a container.

Figures

• Plain figures: Plain figures are all about flat 2-dimensional shapes such as circle, rectangle, etc.

• Solid Figures: Solid Geometry consists of all 3-dimensional figures like cubes, spheres, etc.

Measuring Plain Figures

SQUARE :-

• Perimeter of Square: 4 X Side SIDE

All sides are equal in a square, therefore

No. of sides= 4

Perimeter= Side X 4

Area of Square: Side X Side

Measuring Plain Figures

TRIANGLE:-

• Area of Triangle: ½ X base X height

• Perimeter of Triangle: Side +Side +Side

Measuring Plain Figures

RECTANGLE:-

• Perimeter of Rectangle: 2(length + breadth)

Opposite sides are equal, hence» Perimeter= Length+ breadth+ » length+ breadth» = 2 (length + breadth)

Measuring Solid Figures

CUBE:-

• Surface Area of Cube: 6 a^2 Number of Faces=6

Area of each face= Side X Side

= a X a

= a^2

Total Area= 6 a^2

• Volume of Cube: Length^3

Volume= Length X Breadth X Height

As the length, breadth and height are

all equal hence

Volume= Length^3

Example

Let a cube have a side measuring 2 cm. Find its area as well as volume.

Side=2 cm

Surface area= 6a^2

= 6(2 X 2)

= 6 X 4

Surface Area= 24 cm^2

2 cm

2 cm

Example (Contd.)

Volume= length^3

Length= 2cm

Volume= 2 X 2 X 2

= 8 cm^3

CUBOID:-

• Surface Area of Cuboid: 2(lb + bh + lh)

Number of Rectangle=6

Area of each rectangle=

length X breadth + length X breadth +

length X height + length X height +

breadth X height + breadth X height

Total Surface Area= 2 (lb + bh + lh)

• Volume of Cuboid= length X breadth X height

Example

Let the dimensions of a cuboid be as follows- l=1 cm, b=2 cm, h=3 cm. Find the total surface area and volume.

Surface Area=2(lb+bh+lh)

= 2(1X2 + 2X3 + 1X3)

= 2(2 + 6 + 3)

= 2(11)

= 22 cm^2

1 cm

3 cm

2 cm

Example (Contd.)

Volume= length X breadth X height

= 1 X 2 X 3

= 6 cm^3

Cylinder:-

• Curved Surface Area: 2πrh When we cut this cylinder along the height

then it will form a rectangle with dimensions: 2πr and h. This is the

area of» the curved surface.» 2πr because, the breadth of the

rectangle= circumference of base’s circle.

r

h

2πr

h

• Total Surface Area: 2πr(r+h) Total Surface Area = Area of 2 circles

+ Curved Surface Area

TSA= πr^2 + πr^2 + 2πrh

TSA= 2πr^2 + 2πrh

Total Surface Area = 2πr(r+h)

• Volume: πr^2h

r

h

Example

Let a cylinder have radius=2 cm and height= 7 cm. Find the TSA, CSA and volume of the same.

Radius=2 cm

Height= 7 cm

CSA= 2πrh

= 2 X 22/7 X 2 X 7

= 88 cm^2

7 cm

2 cm

Example (Contd.)

TSA= 2πr(r+h)

= 2 X 22/7 X 2(2+7)

= 2 X 22/7 X 18

= 113.14 cm^2

Volume= πr^2h

= 22/7 X (2)^2 X 7

= 22/7 X 4 X 7

= 88 cm^3

CONE:

• Curved Surface Area: πrs Radius= r

Slant Height= s

(Cut it along the radius and slant height.)

Area of ABC/ Area of circle with centre C= Arc length of AB of sector ABC/

Circumference of circle with centre C

Area of ABC/πs^2 = 2πr/2πs = r/s ,

Area of sector ABC = r/s X πs^2

Curved Surface Area = πrs

r

s

A B

C

s

• Total Surface Area:

= Area of Base + Area of curved surface

= πr^2 + πrs

= πr(r+s)

• Volume: 1/3 πr^2h

Example

Let the radius of the cone be 7 cm and the slant height be 2 cm. Find it’s CSA, TSA and volume.

CSA= πrs

= 22/7 X 7 X 2

= 44 cm^2

TSA= πr(r+s)

= 22/7 X 7(7+2)

7

2

Example (Contd.)

= 22/7 X 7 X 9

= 198 cm^2

Volume= 1/3 πr^2h

= 1/3 X 22/7 X 49 X 2

= 1/3 X 22 X 14

= 1/3 X 308

= 102.66 cm^3

Review Of FormulaeShapes Perimeter Area Curved

Surface Area

Total Surface Area

Volume

Square 4 X Side Side ^2

Rectangle 2(l + b) Length x Breadth

Triangle Side+side+side

½ X b X h

Cube 12a 6a^2 Length^3

Cuboid 4a +4b+ 4c 2(lb+lh+bh) L x b x h

Cone πrs πr(r+s) 1/3πr^2h

Cylinder 2πrh 2πr(r+h) πr^2h

Group Members

Karan Singh Bora

Amaan Ahmad

Vasu Arora

Tushar Sabhani

Utsav Garg