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 22r 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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