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Basic Mensuration

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This document provides information about mensuration, which is the measurement of geometric figures. It begins with acknowledging the teacher for providing the opportunity to present on this topic. It then defines important terms like perimeter, area, volume, and introduces plain and solid figures. Formulas for calculating the perimeter, area, surface area and volume of squares, rectangles, triangles, cubes, cuboids, cylinders and cones are presented along with examples. The document concludes by listing the group members who prepared the presentation.

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MENSURATION

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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.

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Contents Introduction Quote—Unquote Important Terms Figures Measuring Plain Figures Measuring Solid Figures Review Of Formulae Group Members

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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.

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Basic Mensuration

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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.

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

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

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

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

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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.

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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

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Measuring Plain Figures  TRIANGLE:- • Area of Triangle: ½ X base X height • Perimeter of Triangle: Side +Side +Side

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Measuring Plain Figures  RECTANGLE:- • Perimeter of Rectangle: 2(length + breadth) Opposite sides are equal, hence » Perimeter= Length+ breadth+ » length+ breadth » = 2 (length + breadth)

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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

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• Volume of Cube: Length^3 Volume= Length X Breadth X Height As the length, breadth and height are all equal hence Volume= Length^3

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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 2cm 2 cm

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Example (Contd.) Volume= length^3 Length= 2cm Volume= 2 X 2 X 2 = 8 cm^3

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 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)

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• Volume of Cuboid= length X breadth X height

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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 3cm 2 cm

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Example (Contd.) Volume= length X breadth X height = 1 X 2 X 3 = 6 cm^3

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 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

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• 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

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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 7cm 2 cm

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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

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 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

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• Total Surface Area: = Area of Base + Area of curved surface = πr^2 + πrs = πr(r+s) • Volume: 1/3 πr^2h

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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

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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

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Review Of Formulae Shapes 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

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Group Members  Karan Singh Bora  Amaan Ahmad  Vasu Arora  Tushar Sabhani  Utsav Garg

More Related Content

Basic Mensuration

  • 2. 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.
  • 3. Contents Introduction Quote—Unquote Important Terms Figures Measuring Plain Figures Measuring Solid Figures Review Of Formulae Group Members
  • 4. 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.
  • 6. 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.
  • 7. Important Terms • Solid: A body or geometric figure having three dimensions.
  • 8. Important Terms • Surface Area: The total area of the surface of the three dimensional figure.
  • 9. Important Terms • Perimeter: The continuous line forming the boundary of a geometrical figure.
  • 10. Important Terms • Volume: The total space occupied by an object or the space inside a container.
  • 11. 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.
  • 12. 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
  • 13. Measuring Plain Figures  TRIANGLE:- • Area of Triangle: ½ X base X height • Perimeter of Triangle: Side +Side +Side
  • 14. Measuring Plain Figures  RECTANGLE:- • Perimeter of Rectangle: 2(length + breadth) Opposite sides are equal, hence » Perimeter= Length+ breadth+ » length+ breadth » = 2 (length + breadth)
  • 15. 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
  • 16. • Volume of Cube: Length^3 Volume= Length X Breadth X Height As the length, breadth and height are all equal hence Volume= Length^3
  • 17. 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 2cm 2 cm
  • 18. Example (Contd.) Volume= length^3 Length= 2cm Volume= 2 X 2 X 2 = 8 cm^3
  • 19.  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)
  • 20. • Volume of Cuboid= length X breadth X height
  • 21. 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 3cm 2 cm
  • 22. Example (Contd.) Volume= length X breadth X height = 1 X 2 X 3 = 6 cm^3
  • 23.  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
  • 24. • 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
  • 25. 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 7cm 2 cm
  • 26. 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
  • 27.  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
  • 28. • Total Surface Area: = Area of Base + Area of curved surface = πr^2 + πrs = πr(r+s) • Volume: 1/3 πr^2h
  • 29. 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
  • 30. 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
  • 31. Review Of Formulae Shapes 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
  • 32. Group Members  Karan Singh Bora  Amaan Ahmad  Vasu Arora  Tushar Sabhani  Utsav Garg