Presentation on theme: "“SURFACE AREAS AND VOLUME”"— Presentation transcript:
1“SURFACE AREAS AND VOLUME” PRESENTATION ON“SURFACE AREAS AND VOLUME”(CLASS IX)CREATED BY:AMIT.N.YADAVAMIT GARG (TGT MATHS)(KV JRC BAREILLY)
2Cuboid and CubeIn our day to day life ,we come across objects like a Wooden box, a Match box, a Tea packet ,a Chalk box, a Dice , a Book etc. All these objects are made of six rectangular plane regions. These objects are in the shape of a cuboid. “cuboid is a solid bounded by six rectangular plane regions”
4Surface Area of a Cuboid As we have seen that the surface of a cuboid consists of six rectangular faces. “Surface area of a cuboid equals the sum of the areas of its six rectangular faces” Consider a cuboid whose Length is l cm, Breadth is b cm and Height h cm.
5Area of Top face EFGH = (l * b)cm2 Area of Bottom face ABCD =(l*b)cm2Area of Side face AEHD =(b*h)cm2Area of Side face BFGC = (b*h)cm2Area of Front face ABFE = (l*h)cm2Area of Back face DHGC = (l*h)cm2
6Total surface Area of the Cuboid = sum of the areas of all its six faces= lb+lb+bh+bh+lh+lh= 2lb+ 2bh + 2lh= 2 (lb+bh+lh) cm2.
7Surface Area of a Cube“A cuboid whose length, breadth and height are all equal , is called a cube” Surface area of a cube = 2(a*a + a*a + a*a) = 2(a2 + a2 + a2) = 6 a2
8Lateral Surface Area Of a Cuboid If out of the six faces of a cuboid , we only find the sum of the areas of four faces leaving the bottom and top faces. This sum is called the lateral surface area of the cuboid. L.S.A of the cuboid =Area of face AEHD + Area of face BFGC + Area of face ABFE + Area of face DHGC = b*h + b*h + l*h + l*h = 2bh + 2lh = 2 ( l+ b) * h
9Lateral Surface Area of a Cube L.S. Area of a cube = 2(a*a + a*a) = 2(a2 + a2) = 4a2
10Examples… Example 1) Find the Surface Area of a match box whose length , breadth and height are 16 cm,8cm , and 6 cm respectively.Solution) since match box is in the form of a cuboid.Here l=16 cm , b= 8 cm , h= 6 cmSurface Area of Match Box=2(lb + bh + lh)=2(16*8+8*6+16*6) cm2=2( ) cm2=544cm2
11Example 2) Find the surface Area of a cube whose edge is 11 cm Solution) Here l=11cmsurface area of the given cube = 6l2=6 (11)2 cm2= 6 * 121 Cm2=726 cm2
12Example 3)Three cubes each of side 5 cm are joined end to end Example 3)Three cubes each of side 5 cm are joined end to end. Find the surface area of the resulting cuboid.Solution)The dimension of the cuboid so formed are as underl=15cm , b=5cm , h=5cmSurface Area of cuboid= 2(lb + bh + lh)= 2( ) cm2= 350 cm2
13.Example 4) A swimming pool is 20m in length ,15m in breadth and 4m in depth. Find the cost of cementing its floor and walls at the rate of Rs 20 per square metre. Solution) we have l=20m , b=15m , h=4m Area of the four walls = 2(l + b) * h = 2( ) * 4 m2 = 280 m2 Area of the floor of the swimming pool= l * b =(20*15) m2 = 300 m2
14Continue…Total Area to be cemented = ( ) m2 = 580 m2 Cost of cementing of 1 m2 = Rs 20 Cost of cementing the floor and the walls = Rs(20 * 580) = Rs 11600
15Questions for practices… Question 1)The dimensions of the cuboid are in the ratio 1:2:3 and its total surface area is 88m2. Find the dimension.Question 2) The sum of length breadth and depth of a cuboid is 19cm and the length of its diagonal is 11cm. Find the surface area of the cuboid.Question 3) Find the lateral surface area and total surface area of a cube of edge 10cm.Question 4) Each edge of a cube is increased by 50%. Find the percentage increase in the surface area of the cube.
16Consider a right circular cylinder of radius “r” Surface Area of Right circular Cylinder=Consider a right circular cylinder of radius “r”and height “h”Area of the lateral surface of the cylinder=Area of the rectangle= l * b= 2Πr * h= 2 Π r h square unitsb=hl=2Πr
17Surface area of cylinder = Area of rectangle= 2 π r h Other method of Finding Surface area of cylinder with the help of paperrhL=2πrB=hSurface area of cylinder = Area of rectangle= 2 π r h
18Thus , for a cylinder of radius “r” and height “h” , we haveL.S.A = 2 Π r h square unitsEach base Area = Π r2Total Surface Area = (2 Π r h + 2Π r2 )= 2 Π r (h + r ) Square units
19Outer Curved Surface area of cylinder Activity -: Keep bangles of same radius one over another. It will form a cylinder.hrrFormation of Cylinder by banglesCircumference of circle = 2 π rIt is the area covered by the outer surface of a cylinder.Circumference of circle = 2 π rArea covered by cylinder = Surface area of cylinder= (2 π r) x( h)
20Total Surface area of a solid cylinder circular surfacesCurved surface=area of two circular surfacesArea of curved surface += (2 π r) x( h) + 2 π r2= 2 π r ( h+ r)
21Examples… Example1) The curved surface area of a right circular cylinder of height 14 cm is 88 cm2.Findthe diameter of the base of the cylinder.Solution)Let r be radius and h be the height of the cylinder.2 π r h = 88 and h = 142 * 22/7 * r * 14 = 8888 r = 88r = 1Diameter of the base = 2r = 2 cm
22Hence, radius of the cylinder is 7 cm. 44 cm Example 2) A rectangular sheet of paper 44 cm * 18 cm is rolled along its length and a cylinder is formed. Find the radius of the cylinder.Solution) Let r be the radius of the base and h be the height. Then , h = 18 cm2 π r = 44r = 7cmHence, radius of the cylinder is 7 cm.44 cm18 cm18 cm
23Example 3) The ratio between the curved surface area and the total surface area of a right circular cylinder is 1:2 . Find the ratio between the height and the radius of the cylinder .Solution)Let h be the height and r be the radius and of the cylinder.2 π r h / 2 π r h + 2 π r2 = 1/22 π r h / 2 π r (h + r) = ½h / h + r = ½2h = h + rh = rh:r = 1 :1
24Questions for practices… Question 1) Curved surface area of a right circular cylinder is4.4m2 . If the radius of the base of the cylinder is 0.7 m , findits height.Question 2) In a hot water heating system , there is a cylindrical pipe of length 28m and diameter 5cm.Find the total radiating surface in the system.Question 3) A cylindrical pillar is 50cm in diameter and 3.5m in height. Find the cost of painting the curved surface of the pillar at the rate of per m2 .Question 4)It is required to make a closed cylindrical tank of height 1m and base diameter 140cm from a metal sheet. How many square meters of the sheet are required for the same?
26Curved surface Area of the cone = Area of the sector VAB lABCurved surface Area of the cone = Area of the sector VAB= ½ * ( arc length * radius)= ½ * 2 π r * l= π r lC.S.R = ½ * (circumference of base * slant height)
27Curved Surface Area of a Cone l=2rCurved Surface Area of a Cone = 1 / 2 * l * 2 Π r= Π r lTotal Surface Area of a cone = Π r l + Π r2= Π r ( l + r)
28Examples… Example 1)The diameter of a cone is 14cm and its slant height is 9 cm. Find the area of itscurved surface .Solution) S = Π r lHere , r = 14/2 cm and l = 9 cms = 22/7 * 7 * 9 cm2 = 198 cm2Example 2) Find the total surface area of a cone , ifits slant height is 9 m and the radius of its base is12m.Solution) S = Π r l + Π r2S=Π r ( l + r)S = ( 22/7 * 12 * (12 + 9)) m2= 792 m2
29Example 3) The radius of a cone is 3 cm and vertical height is 4 cm Example 3) The radius of a cone is 3 cm and vertical height is 4 cm. Find the area of the curved surface.Solution)we have r = 3 cm and h = 4 cml2 = r2 + h2l2 = 3*3+ 4*4l2 = 5 cmArea of the curved surface = S = Π r l=22/7 * 3 * 5= 47.14cm24cm3cm
30Questions for practices… Question 1) Find the curved surface area of a cone , if its slant height is 60 cm and the radius of its base is 21 cm.Question 2) The radius of a cone is 5cm and vertical height is 12cm. Find the area of the curved surface.Question 3)The radius of a cone is 7cm. And area of curved surface is 176cm2 .Find the slant height.
31Surface Area of a Sphere Surface Area of a Sphere = 4 Π r2Curved Surface Area of a Hemi Sphere = 2 Π r2Total Surface Area of a Hemi Sphere = 2 Π r2 + Π r2= 3 Π r2
32Examples… Example 1) Find the surface area of a sphere of radius 7cm. Solution) s = 4 Π r2Here , r = 7cms = 4 * 22/7 * 7 * 7 cm2 = 616 cm2Example 2) Find the surface area and total surface area of a hemisphere of radius 21cm.Solution) S = 2 Π r 2 and s1 = 3 Π r2Here, r = 21s= 2 * 22/7 * 21 * 21 cm2s= 2772 cm2 s1= 4158 cm2
33Questions for practices… Question1) Find the surface area of a sphere of radius(i) 10.5 cm (ii) 5.6 cm (iii) 14 cmQuestion 2) Find the surface area of a sphere of diameter.(i)14 cm (ii) 21cm (iii)3.5cm
34Continue…Question 3) Find the total surface area of a hemi sphere and a solid hemisphere each of radius 10cm .Question 4) The surface area of a sphere is 5544cm2 , find its diameter.
35Volume of a Cuboid Volume of a Cuboid = Base area * Height = length * breadth * Height
36Examples…Example 1) The volume of a cuboid is 440 cm3 and the area of its base is 88cm2. Find the breadth of the tank if its length and the depth are respectively 2.5m and 10m.Solution)Volume = 440 cm3 and Area of the base = 88cm2Height = Volume / Area of the base= 440/ 88 cm = 5 cm
37Questions for practices… Question 1) A cuboidal water tank is 6m long , 5m wide and 4.5 m deep. How many litre of water can it hold?Question 2) A cubical vessel is 10 m long and 8m wide. How high must it be made to hold 380 cubic metres of a liquid ?
38VOLUME OF CUBE Area of base (square) = a2 Height of cube = a Volume of cube = Area of base x height= a2 x a = a3(unit)3
39Examples….Example 1) How many 3 metre cubes can be cut from a cuboid measuring 18 m * 12 m * 9 m.Solution) Edge of each cube = 3mVolume of each cube = (edge)3 = 3*3*3 m3= 27m3Volume of the cuboid = (18 * 12 * 9) m3= m3Number of cube= volume of the cuboid/volume of the cube= / 27= 72
40Volume of cylinder Volume of cylinder = Area of base x vertical height = π r2 * h
41Examples…Example 1) Find the volume of a right circular cylinder , if the radius (r) of its base and height(h) are 7cm and 15cm respectively.Solution)volume of cylinder = π r2 hHere r = 7cm and h = 15cmVolume of the cylinder = 22/7 * 7 * 7 * 15 cm3= 2310 cm3
42Example 2) The area of the base of a right circular cylinder is 154cm2 and its height is 15cm . Find the volume of the cylinder.Solution)volume of a cylinder= area of the base * heightHere , area of the base = 154cm and height= 15cm= (154 * 15) cm3
43Questions for practices… Question 1 ) Find the volume of a right circular cylinder , if the radius (r) of its base and height (h) are 7cm and 15cm respectively.Question 2) The volume of a cylinder is 448 π cm3. Find its lateral surface area and total surface area.Question 3) A well with 10m inside diameter is dug 14m deep. Earth taken out of it is spread all around to a width of 5m to form an embankment. Find the height of embankment.
45Volume of a Cone 3( volume of cone) = volume of cylinder hhHere the vertical height and radius of cylinder & cone are same.rr3( volume of cone) = volume of cylinder3( V ) = π r2hV = 1/3 π r2h
46If both cylinder and cone have same height and radius then volume of a cylinder is three times the volume of a coneVolume = 3VVolume =V
47Mr. 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.
48ExamplesExample 1) A conical tank is 3m deep and its circular top has radius 1.75m. Find the capacity of the tank in kilolitres.Solution)we have , r= 1.75 m , h= 3mCapacity of the tank = 1/3 π r2hCapacity of the tank= 1/3 * 22/7 *1.75 * 1.75 * 3 m3 Capacity of the tank = m3= kilolitre( 1m3 = 1 kilolitre)
49Example 2) The height and the slant height of a cone are 21 cm and 28 cm respectively. Find the volume of the cone.Solution )l2 = r2 + h2r = 7√ 7 cm.volume of the cone = 1/3 π r2h=1/3 * 22/7 *7√7 *√7 *21 cm3= 7546cm3
50Questions for practices… Question 1 ) Find the volume of a right circular cone 1.02m high, if the radius of its base is 28cm.Question 2) The area of the base of a right circular cone is 314cm2 and its height is 15cm. Find the volume of the cone.Question 3) A semi-circular sheet of metal of diameter 28cm is bent into an open conical cup. Find the depth and capacity of cup.
51Comparison of Area and volume of different geometrical figures Surface area6a22π rhπ r l4 π r2Volumea3π r2h1/3π r2h4/3 π r3
52Volume of a Sphere V = 4/3 π r3 volume of Sphere =4( volume of cone) h=rrHere the vertical height and radius of cone are same as radius of sphere.volume of Sphere =4( volume of cone)V = 4( 1/3πr2h ) = 4( 1/3πr3 )V = 4/3 π r3
53If 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.V1rrrV=1/3 πr2hIf h = r thenV=1/3 πr3V1 = 4V = 4(1/3 πr3)= 4/3 πr3
54Examples…. Example 1) Find the volume of a sphere of radius 7cm. Solution) V= 4/3 πr3Here, r = 7cmV = 4/3 * 22/7 * 7 * 7 * 7 cm3= cm3Example 2) A hemispherical bowl has a radius of 3.5 cm. What would be the volume of water it would contain?Solution)The volume of water the bowl can contain = 2/3 πr3= 2/3 *22/7 * 3.5*3.5*3.5 cm3= 89.8 cm3
55Questions for practices… Question 1) A hemispherical bowl is made of steel 0.5cm thick. The inside radius of the bowl is 4cm. Find the volume of steel used in making the bowl.Question 2) The volume of the two sphere are in the ratio 64:27. Find the difference of their surface areas , if the sum of their radii is 7.Question 3) A solid sphere of radius 3cm is melted and then cast into small spherical balls each of diameter 0.6 cm. Find the number of balls thus obtained.
56Volume of a Cylinder= Π r2 h Volume of a Right Circular Cone =1/3 π r2hVolume of a sphere = 4/3 π r3Volume of a Hemi sphere = 2/3 π r3