Basic Mensuration
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Transcript of Basic 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.
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
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)
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
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