Mensuration

Deals with the understanding of surface area and volume of solid geometrical figures-cube, cuboid, cylinder, cone and sphere

1. M E N S U R A T I O N- sk STD X M A H A R A S H T R A S T A T E B O A R D O F E D U C A T I O N , M U M B A I

2. MENSURATION Mensuration  is a branch of  mathematics  which deals with the surface area and volume of solid, plane and geometrical figures.

3. MENSURATION The  area  of a figure is the number of unit squares that cover the surface of a closed figure.  Area  is measured in square units such as square centimetres, square feet, square inches, etc.

4. MENSURATION In  math ,  volume  can be defined as the 3-dimensional space enclosed by a boundary or occupied by an object. ... Here, for example, the  volume  of the cuboid or rectangular prism, with unit cubes has been determined in cubic units.

5. MENSURATION

6. MENSURATION CURVED SURFACE AREA (LATERAL SURFACE AREA) TOTAL SURFACE AREA AND THE AREA OF TOP AND BOTTOM SURFACES CURVED SURFACE AREA (LATERAL SURFACE AREA)

7. MENSURATION- surface area CUBE A cube is a three-dimensional solid (has length, breadth, height) object bounded by six square faces, with three meeting at each vertex. The cube is the only regular hexahedron and is one of the five Platonic solids . (solids having regular faces- refer adjacent figure) Area of each surface = (side) 2 Lateral surface area = 4 x (side) 2 Total surface area = 6 x (side) 2

8. MENSURATION- surface area CUBOID A cuboid is a convex polyhedron bounded by six quadrilateral faces, Having dimensions l, b and h Area of a blue surface = l x h Area of a pink surface = b x h area of a green surface = l x b Lateral surface area = 2 ( lh + bh ) =2 h (l +b) Total surface area =2 ( lh + hb + lb ) = 2( lb + bh +hl)

9. MENSURATION- surface area h CYLINDER A cylinder has traditionally been a three-dimensional solid. It is the idealized version of a solid physical tin can having lids on top and bottom. Dimensions h, r L= Circumference of a circle     Area of rectangle= l x h = 2 h Lateral surface area = 2 h Total surface area =2 h + 2 r 2 =2 (h + r)   L

10. MENSURATION- surface area CONE A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex . Dimensions: h, r, l Curved surface area = l Total surface area = l + = (r +l) Remember : l 2 = r 2 + h 2    

11. MENSURATION- surface area SPHERE AND HEMISPHERE A sphere is a geometrical object in three-dimensional space that is the surface of a ball . Dimension: r Curved surface area = 4 2   Curved surface area = 2 2 Total surface area = 3 2  

12. MENSURATION-volume In  math ,  volume  can be defined as the 3-dimensional space enclosed by a boundary or occupied by an object. ... GENERAL FORMULA: volume = A(BASE) x HEIGHT VOLUME OF A CUBE = A(SQUARE) x H V (CUBE ) = side x side x H V (cube ) = (side) 3 ( H = side)

13. MENSURATION-volume GENERAL FORMULA: volume = A(BASE) x HEIGHT VOLUME OF A CUBOID =A(rectangle) x H V(CUBOID) = L x B x H VOLUME OF A CYLINDER = A(CIRCLE) x H V(CYLINDER) = 2 x h  

14. MENSURATION-volume cone The volume of a cone means the third part of the volume of a cylinder having the same base and the same height. It takes three  cones  to fill up a  cylinder .

15. MENSURATION-volume sphere The  sphere volume  is 2/3 of the  volume  of a  cylinder with  the same radius and height equal  to  the diameter.  V(CYLINDER) = 2 x h V(SPHERE) = x 2 x 2 r V(SPHERE) = x 3  

16. MENSURATION-volume hemisphere V(CYLINDER) = 2 x h V(SPHERE ) = x 2 x 2 r V(SPHERE) = x 3 V(HEMISPHERE) = x 3 =   x 3  

17. Formulae to find SURFACE AREA AND VOLUME 3D –SOLID FIGURE DIMENSIONS SURFACE AREA VOLUME     CURVED TOTAL   CUBE Side 4 x side 2 6 x side 2 Side 3 CUBOID l,b,h 2 h (l +b)   2( lb + bh +hl)   L x b x h CYLINDER r,h 2 h 2 (r + h) 2 x h CONE r,h,l l (r +l) x 2 h SPHERE r 2 2 x 3 HEMISPHERE r 2 2 x 3 3D –SOLID FIGURE DIMENSIONS SURFACE AREA VOLUME     CURVED TOTAL   CUBE Side 4 x side 2 6 x side 2 Side 3 CUBOID l,b,h 2 h (l +b)   2( lb + bh +hl)   L x b x h CYLINDER r,h CONE r,h,l SPHERE r HEMISPHERE r

18. APPLICATION SURFACE AREA AND VOLUME For a cone: r =1.5cm, h = 5cm To find: volume of a cone Formula: v(cone) = x 2 H = x 3.14 x 1.5 x 1.5 x 5 = x x x 5 =11.775 cubic cm (157 X 75)   Find the volume of a cone if the radius of its base is 1.5cm and its perpendicular height is 5 cm.

19. APPLICATION SURFACE AREA AND VOLUME Find the surface area of a ball. 42

20. APPLICATION SURFACE AREA AND VOLUME Cylinder: r =5 cm h = 40 cm To find: total surface area of the cylinder.   Total surface area (cylinder )= 2 (h + r) = 2 x x 5 (5+40) = 2 x x 5 x 45 cubic cm = 1413 cubic cm    

21. APPLICATION SURFACE AREA AND VOLUME TOTAL S AREA = C S (HEMISPHERE) +C S (CYLINDER) + C S (CONE) = 2 2 + 2 h c + l = r (2r + 2h c + l) Calculation for slant height (l) : l 2 = r 2 + h cone 2 =3 2 + 4 2 = 9+16 = 25 L = 5 total area= r(2X3 + 2 X 40 + 5) = X 3 (91) = 22 X 3 X 13 sq. cm   =858 sq. cm     Hemisphere : R = 3 cm, Cylinder : r =3 cm H c = 40 cm Cone : r =3 cm, h cone = 4 cm, l=? To find: surface area of the toy  

22. APPLICATION SURFACE AREA AND VOLUME Cuboid: l= 44 cm, b = 21cm, h = 12cm Cone r = ?, h =24 cm Volume of cuboid = volume of cone lXbXh = x 2 h 44X21X12 = X X 2 X 24 8 = r 2 21 X 21 = 2 21 = radius of the base of a cone.  

23. APPLICATION SURFACE AREA AND VOLUME Cylinder: 2 = 100cm 2 , h = 3 cm Cone : H=? V(cylinder +cone) = 500 cm 3   Total height of the figure= 6+3 = 9cm

24. APPLICATION SURFACE AREA AND VOLUME In a cylindrical glass, diameter = 14cm and h=30 cm, containing water, a metal sphere of diameter 2 cm is immersed. Find the volume of the water. Cylinder: d 1 = 14cm, h = 30 cm, sphere: d 2 = 2 cm. To find: volume of water in the cylinder Volume of water = volume of cylinder - volume of sphere = 2 x h - x 3 = (49X30 - X 1) = ( 1470 - 1.33 ) = (1468.67) cubic cm   In the cylinder

25. APPLICATION -SURFACE AREA AND VOLUME Find the volume and the surface area of the toy shown. Given: for a cone: r = 3 cm, h 1 = 4 cm for a hemisphere: r = 3 cm To find total volume and total surface area Total volume= v(cone) + v(hemisphere) = x 2 h 1 + x 3 = 2 ( h 1 + ) =3.14 X9(1.33+2 ) ………( note: h 1 =4cm) =28.26(3.33)= 94.1 cm 3   Length of lateral surface: L 2 = r 2 + h 1 2 , l 2 = 4 2 + 3 2 , l = 5 Total surface are = surface area of cone + surface area of hemisphere = l + 2 2 = (I + 2r) = 3.14 X3(5+ 6) = 9.42(11) = 103.62 cm 2  

26. APPLICATION SURFACE AREA AND VOLUME Using information given in the figure, find how many jugs of water can the cylindrical pot hold? (measurements are in cm). 10 3.5 10 R=7

27. APPLICATION SURFACE AREA AND VOLUME The radius of a tablet in a cylindrical wrapper is 7 mm and its thickness is 5 mm. if the height of the wrapper is 10cm and diameter is 14 mm then find the number of tablets in the wrapper Wrapper (cylindrical): diameter: 14mm, height 10 cm = 100 mm Tablets(cylindrical) : r = 7 mm, height=5 mm

28. APPLICATION -SURFACE AREA AND VOLUME Find the ratio of the volumes of a cylinder and a cone having equal radius and equal height The radii of two cylinders are in the ratio 2:3 and their heights are 5:3. Find the ratio of their volumes ( 2 x h) 1 : ( 2 x h) 2 X 2X 2X5 : X3 X 3 X3 20: 27   Find the ratio of the volumes of a cylinder, a cone and hemisphere having equal radius and equal height 2 x h : x 2 h : x 3 3 : 1 : 2  

29. REVISION How much oil a cuboid can, having dimensions 20cm, 20 cm and 30 cm contain? ( 1 litre = 1000 cm 3 ) How much cloth is needed to stitch a conical cap having radius of its base as 10 cm and slant height 21 cm. How many solid cylinders of radius 10cm and height 6 cm can be made by melting a solid sphere of radius 30 cm? Find the curved surface area of a cone of radius 7 cm and height 24cm.

30. REVISION Find the volume of a cube having length of side 6 cm. In a cylinder if radius is halved and height is doubled then volume will be – same / double/ halved/four times? The curved surface area of a cylinder is 440 cm 2 and its radius is 5 cm then find its height.

31. T H A N K Y O U