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					       Surface Area & Volume

                   OBJECTIVES
• Slices of 3-dimensional figures
• Nets of 3-D solids
• Lateral areas, surface areas,volumes of solids
• Properties of congruent & similar solids
          3-dimensional figures
You will use different math websites for
  visualizations of 3-D figures.
• Take time to use these sites to help explain the
  material.
• Refer back to them for formulas, detailed
  diagrams, and rotations
• If the links are blocked by your firewall, copy the
  URL & paste it in the address on your homepage
• Bookmark them or add to ‘favorites’ while you’re
  in this course.
Websites for 3-dimensional geometry
    http://www.mathsisfun.com/platonic_solids.html

    http://www.mathsisfun.com/geometry/index.html

http://mathforum.org/alejandre/workshops/polyhedra.html

http://mathforum.org/library/drmath/sets/high_geom.html

http://www.houseof3d.com/pete/applets/graph/index.html
              Nets & surface areas
• Nets are ‘flattened’ 3-D figures– it’s the pattern to
  make the shape by folding.
• Refer to the text for vocabulary and diagrams as
  well as the websites
• Add areas of all non-overlapping sections of the net
  for each polygon for surface area of figure.
    http://www.mathsisfun.com/platonic_solids.html
      Surface area of prisms & cylinders
  • Refer to the websites and your text for diagrams
    and vocabulary: bases, lateral edges, altitude,
    prism & cylinder
  • Lateral area is the surface area minus the bases
  • Surface area includes both bases
    Right cylinder                  Regular prism
L.A. = Ph = 2 π r h                  L.A. = P h
S.A. = Ph+2π r2 = 2 π r h + 2π r2    S.A. = P h + 2B
  Surface areas of pyramids & cones
 http://www.mathsisfun.com/geometry/pyramids.html

• Refer to the websites and your text for diagrams
  and vocabulary: base, vertex, lateral faces & edges,
  altitude, and slant height.
• Refer to these resources for formulas
       Volume of prisms & cylinders

• Right prisms:
  Volume = B • h, where B = area of base & h = height

• Right cylinder:
  Volume = B • h = π r 2 h
       Volume of pyramids & cones
• Volume of Right pyramids:

         1
      V = B h, where B = area of base
         3

• Volume of right cone:
     V = 1 B h = 1 h r 2
         3         3
Cavalarieri’s Principle: 2 solids of same height &
  same cross-sectional area have the same volume.
    Surface area & volume of spheres
• Refer to text for vocabulary

• Surface Area = 4 π r 2

           4
• Volume =    r3
           3
        Congruent & similar solids
2 solids are congruent if:
     corr 's  & corr edges 
     areas of corr faces  & volumes 


2 solids are similar:
   if ratio of 2 solids are a : b, then
        surface area ratio – a 2 : b2
        volume ratio – a 3 : b3
          1.2 SURFACE AREA OF
            A CUBOID & A CUBE ~
1 SURFACE AREA OF A CUBOID ~
The total surface area (TSA) of a cuboids is the
sum of the areas of its six faces. That is:
 Example 1: Find the total surface area of a
cuboids with dimensions 8 cm by 6 cm by 5 cm.

                 Solution:
              Surface area of a cube ~
To derive the formula of the surface area of a cube, start with a
    cube as shown below and call the length of one side a:
   In order to make a cube like the one
    shown above, you basically use the
         following cube template:




Looking at the cube template, it is easy to see that the cube
has six sides and each side is a square
The area of one square is a × a = a2
               Example~2:
Find the surface area if the length of one side is
                     5 cm

                    SOL ~
             Surface area = 6 × a2

             Surface area = 6 × 42

           Surface area = 6 × 4 × 4

            Surface area = 96 cm2
SURFACE AREA OF A RIGHT CIRCULAR
           CYLENDER ~
Curved Surface Area of a Cylinder
             Recall:
           Total Surface Area of a Cylinder ~

             Consider a cylinder of radius r and height h.
The total surface area (TSA) includes the area of the circular top and
          base, as well as the curved surface area (CSA)
The total surface area (TSA) of a cylinder with radius r and height h is
given by
Example 3 ~ Find the total surface area of a cylindrical
        tin of radius 17 cm and height 3 cm.
                       Solution:
SURFACE AREA OF A RIGHT
   CIRCULAR CONE ~
  Total Surface Area of a Cone ~
            Example 4~



Solution:
surface area = 4pr2

Example 5 :~ if 'r' = 5 for a given sphere, and p =
3.14, then the surface area of the sphere is:
surface area = 4pr2
      = 4 × 3.14 × 52
      = 314
               Volume of a Cuboid
A cuboid with length l units, width w units and height h units has
               a volume ofV cubic units given by
                              V = lwh
             Example 5
Find the volume of a brick 30 cm by
          25 cm by 10 cm.
              Solution:
     Volume of a Cylinder ~
 A cylinder with radius r units and
         height h units has
a volume of V cubic units given by
Example 6 ~ Find the volume of
   a cylindrical canister with
 radius 7 cm and height 12 cm.
            Solution:
      Volume of a Cone ~
The volume of a cone is given by
Example 42


 Solution:
              Volume of a Sphere ~
If four points on the surface of a sphere are joined to the
  centre of the sphere, then a pyramid of perpendicular
height r is formed, as shown in the diagram. Consider the
solid sphere to be built with a large number of such solid
pyramids that have a very small base which represents a
      small portion of the surface area of a sphere
 Surface area and volume of different Geometrical Figures




Cube         Parallelopipe
                               Cylinder      Cone
             d
              Faces of cube

    face




                face
    face




                       1
                       2   3

                  Dice (Pasa)



Total faces = 6 ( Here three faces are visible)
                                Faces of Parallelopiped


          Face
Face


            Face




       Total faces = 6 ( Here only three faces are
       visible.)

                                                          Book

                         Bric
                         k
                      Cores
               Core
               s




Total cores = 12 ( Here only 9 cores are visible)



         Note Same is in the case in parallelopiped.
                                 Surface area
           Cube                                 Parallelopiped




                                                                   c
                  a
                                                                             b
                                                      a
                          a                                      Click to see the faces of
                                                                 parallelopiped.
            a
   (Here all the faces are square)     (Here all the faces are rectangular)



Surface area = Area of all six        Surface area = Area of all six
faces                                 faces
                      = 6a2                     = 2(axb + bxc +cxa)
                     Volume of Parallelopiped   Click to animate




                                            c

      b
                        a                  b

Area of base (square) = a x b

Height of cube = c
Volume of cube = Area of base x
height x b) x c
 = (a
                           Volume of Cube   Click to see




    a                                 a

                 a


Area of base (square) =
a2
Height of cube = a
Volume of cube = Area of base x height
=       a2 x a       = a3(unit)   3
        Outer Curved Surface area of cylinder               Click to animate




                                                           Activity -: Keep
                                                           bangles        of
          r                                  h
                                     r                     same       radius
                                                           one          over
                                                           another. It will
Circumference of          Formation of                     form a cylinder.
circle = 2 π r            Cylinder by bangles

       It is the area covered by the outer surface of a cylinder.
   Circumference of circle = 2 π r
   Area covered by cylinder = Surface area of of cylinder = (2 π r)
    Total Surface area of a solid cylinder




                                    Curved
     circular                       surface
     surface
     s



= Area of curved surface + area of two circular surfaces

      =(2 π r) x( h) + 2 π r2

      = 2 π r( h+ r)
    Other method of Finding Surface area of cylinder with the help of
paper
                  r



                        h




                                                   h



                         2πr
        Surface area of cylinder = Area of rectangle= 2 πrh
               Volume of cylinder




                               r


                                        h




Volume of cylinder = Area of base x vertical height
= π r2 xh
       Cone




              h
Base
                  r
                Volume of a Cone              Click to See
                                              the
                                              experiment




      h                                   h
          Here the vertical height and
          radius of cylinder & cone are
          same.                                     r
r
    3( volume of cone) = volume of cylinder
    3( V )        = π r2 h
    V = 1/3 π r2h
if both cylinder and cone have same height and radius then volume of a cylinder is three
times the volume of a cone ,




               Volume = 3V                         Volume =V
Mr. Mohan has only a little jar of juice he wants to
distribute it to his three friends. This time he choose the
cone shaped glass so that quantity of juice seem to
appreciable.
                         Surface area of cone

                                                                    l




                                                          2πr
             l



                                                                l




Area of a circle having sector (circumference) 2π l = π l2πr
                                                          2


Area of circle having circumference 1 = π l 2/ 2 π l
         Comparison of Area and volume of different geometrical figures




Surface   6a2         2π rh          πrl            4 π r2
area




Volume    a3          π r 2h         1/3π r2h       4/3 π r3
               Area and volume of different geometrical figures


                                                   r
                                   r
                                                                  r/√2
    r                        r                         l=2
                                                       r
Surface   6r2            2π r2         2π r2            2 π r2
area
          =2 π r2
               (about)

Volume    r3             3.14 r3       0.57π r3         0.47π r3
    Total surface Area and volume of different geometrical figures and
nature
                                                    r
                                  r
                                                                        r
                             r              22r
    1.44r                                               l=3
                                                        r
   Total     4π r2      4π r2          4π r2                  4 π r2
   Surface
   area


   Volume    2.99r3     3.14 r3        2.95 r3                4.18 r3


 So for a given total surface area the volume of sphere is maximum.
 Generally most of the fruits in the nature are spherical in nature
 because it enables them to occupy less space but contains big amount
Think :- Which shape (cone or cylindrical) is better for collecting
resin from the tree




                                                               Click the
             r                          r



                           3r




V= 1/3π r2(3r)                  V= π r2 (3r)
V= π r3
Long but Light in weight        V= 3 π r3
                                Long but Heavy in weight
 Small niddle will
require to stick it             Long niddle
in the tree,so little           will require
harm in tree
                                to stick it in
                                the tree,so
Bottle

          Cone
          shape

         Cylindrical
         shape
If we make a cone having radius and height equal to the radius of
sphere. Then a water filled cone can fill the sphere in 4 times.




                                                       r
            r
        r

     V=1/3 πr2h
                                               V1
     If h = r then
     V=1/3 πr3
                      V1 = 4V = 4(1/3
                      πr3)
                      = 4/3 πr3
               Volume of a Sphere            Click to See
                                             the
                                             experiment




                                        h=
                                        r          r
        Here the vertical height and
        radius of cone are same as
        radius of sphere.
r   4( volume of cone) = volume of Sphere
    4( 1/3πr2h ) = 4( 1/3πr3 ) = V
    V = 4/3 π r3
Made by :- Anukul Saini
           Prashu
           Kushagra roy
           Kuldeep
           Rishab Rawat
           Vishal Lohan


                          U.C. Pandey R.C.Rauthan, G.C.Kandpal
Thanking you

				
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Description: maths project surface area volume