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CREATED BY: AMIT.N.YADAV AMIT GARG (TGT MATHS) (KV JRC BAREILLY) PRESENTATION ON SURFACE AREAS AND VOLUME (CLASS IX) Slide 2 Cuboid and Cube In 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 Slide 3 Slide 4 Surface 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. Slide 5 Area of Top face EFGH = (l * b)cm 2 Area of Bottom face ABCD =(l*b)cm 2 Area of Side face AEHD =(b*h)cm 2 Area of Side face BFGC = (b*h)cm 2 Area of Front face ABFE = (l*h)cm 2 Area of Back face DHGC = (l*h)cm 2 Slide 6 . Total 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) cm 2 Slide 7 Surface 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(a 2 + a 2 + a 2 ) = 6 a 2 Slide 8 Lateral 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 Slide 9 Lateral Surface Area of a Cube L.S. Area of a cube = 2(a*a + a*a) = 2(a 2 + a 2 ) = 4a 2 Slide 10 Example 1) Find the Surface Area of a match box whose length, breadth and height are 16 cm,8 cm, and 6 cm respectively. Solution) since match box is in the form of a cuboid. Here l=16 cm, b= 8 cm, h= 6 cm Surface Area of Match Box=2(lb + bh + lh) =2(16*8+8*6+16*6) cm 2 =2(128+48+96) cm 2 =544cm 2 Examples Slide 11 Example 2) Find the surface Area of a cube whose edge is 11 cm Solution) Here l=11cm surface area of the given cube = 6l 2 =6 (11) 2 cm 2 = 6 * 121 Cm 2 =726 cm 2 Slide 12 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 under l=15cm, b=5cm, h=5cm Surface Area of cuboid= 2(lb + bh + lh) = 2(75+25+75) cm 2 = 350 cm 2 Slide 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(20 + 15) * 4 m 2 = 280 m 2 Area of the floor of the swimming pool= l * b =(20*15) m 2 = 300 m 2 Slide 14 Continue Total Area to be cemented = (280 + 300) m 2 = 580 m 2 Cost of cementing of 1 m 2 = Rs 20 Cost of cementing the floor and the walls = Rs(20 * 580) = Rs 11600 Slide 15 Questions for practices Question 1)The dimensions of the cuboid are in the ratio 1:2:3 and its total surface area is 88m 2. 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. Slide 16 = 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 = 2r * h = 2 r h square units Surface Area of Right circular Cylinder l=2r b=h Slide 17 L=2 r B=h r h Surface area of cylinder = Area of rectangle= 2 r h Other method of Finding Surface area of cylinder with the help of paper Slide 18 Thus, for a cylinder of radius r and height h, we have L.S.A = 2 r h square units Each base Area = r 2 Total Surface Area = (2 r h + 2 r 2 ) = 2 r (h + r ) Square units Slide 19 r h Activity -: Keep bangles of same radius one over another. It will form a cylinder. It is the area covered by the outer surface of a cylinder. Formation of Cylinder by bangles Circumference of circle = 2 r r Outer Curved Surface area of cylinder Circumference of circle = 2 r Area covered by cylinder = Surface area of cylinder = (2 r) x( h) Slide 20 = (2 r) x( h) + 2 r 2 Curved surface Area of curved surface + area of two circular surfaces = circular surfaces = 2 r ( h+ r) Total Surface area of a solid cylinder Slide 21 Example1) The curved surface area of a right circular cylinder of height 14 cm is 88 cm 2.Find the diameter of the base of the cylinder. Solution) Let r be radius and h be the height of the cylinder. 2 r h = 88 and h = 14 2 * 22/7 * r * 14 = 88 88 r = 88 r = 1 Diameter of the base = 2r = 2 cm Examples Slide 22 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 cm 2 r = 44 r = 7cm Hence, radius of the cylinder is 7 cm. 44 cm 18 cm Slide 23 Example 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 r h / 2 r h + 2 r 2 = 1/2 2 r h / 2 r (h + r) = h / h + r = 2h = h + r h = r h:r = 1 :1 Slide 24 Question 1) Curved surface area of a right circular cylinder is 4.4m 2. If the radius of the base of the cylinder is 0.7 m, find its 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 12.50 per m 2. 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? Questions for practices Slide 25 2r2r l Surface area of a Right Circular cone Slide 26 2r2r l Curved surface Area of the cone = Area of the sector VAB = * ( arc length * radius) = * 2 r * l = r l C.S.R = * (circumference of base * slant height) V A B Slide 27 r l=2r Curved Surface Area of a Cone Curved Surface Area of a Cone = 1 / 2 * l * 2 r = r l Total Surface Area of a cone = r l + r 2 = r ( l + r) Slide 28 Example 1)The diameter of a cone is 14cm and its slant height is 9 cm. Find the area of its curved surface. Solution) S = r l Here, r = 14/2 cm and l = 9 cm s = 22/7 * 7 * 9 cm 2 = 198 cm 2 Example 2) Find the total surface area of a cone, if its slant height is 9 m and the radius of its base is12m. Solution) S = r l + r 2 S= r ( l + r) S = ( 22/7 * 12 * (12 + 9)) m 2 = 792 m 2 Examples Slide 29 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 cm l 2 = r 2 + h 2 l 2 = 3*3+ 4*4 l 2 = 5 cm Area of the curved surface = S = r l =22/7 * 3 * 5 = 47.14cm 2 4cm 3cm Slide 30 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 176cm 2.Find the slant height. Questions for practices Slide 31 Surface Area of a Sphere = 4 r 2 Curved Surface Area of a Hemi Sphere = 2 r 2 Total Surface Area of a Hemi Sphere = 2 r 2 + r 2 = 3 r 2 Surface Area of a Sphere r 2r Slide 32 Example 1) Find the surface area of a sphere of radius 7cm. Solution) s = 4 r 2 Here, r = 7cm s = 4 * 22/7 * 7 * 7 cm 2 = 616 cm 2 Example 2) Find the surface area and total surface area of a hemisphere of radius 21cm. Solution) S = 2 r 2 and s1 = 3 r 2 Here, r = 21 s= 2 * 22/7 * 21 * 21 cm 2 s= 2772 cm 2 s1= 4158 cm 2 Examples Slide 33 Question 2) Find the surface area of a sphere of diameter. (i)14 cm (ii) 21cm (iii)3.5cm Questions for practices Question1) Find the surface area of a sphere of radius (i) 10.5 cm (ii) 5.6 cm (iii) 14 cm Slide 34 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 5544cm 2, find its diameter. Continue Slide 35 Volume of a Cuboid = Base area * Height = length * breadth * Height Volume of a Cuboid b Slide 36 Example 1) The volume of a cuboid is 440 cm 3 and the area of its base is 88cm 2. Find the breadth of the tank if its length and the depth are respectively 2.5m and 10m. Solution) Volume = 440 cm 3 and Area of the base = 88cm 2 Height = Volume / Area of the base = 440/ 88 cm = 5 cm Examples Slide 37 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 ? Questions for practices Slide 38 a a Area of base (square) = a 2 Height of cube = a Volume of cube = Area of base x height = a 2 x a = a 3 a (unit) 3 VOLUME OF CUBE Slide 39 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 = 3m Volume of each cube = (edge) 3 = 3*3*3 m 3 = 27m 3 Volume of the cuboid = (18 * 12 * 9) m 3 = 1944 m 3 Number of cube = volume of the cuboid/volume of the cube = 1944 / 27 = 72 Examples. Slide 40 Volume of cylinder = Area of base x vertical height = r 2 * h r h Volume of cylinder Slide 41 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 = r 2 h Here r = 7cm and h = 15cm Volume of the cylinder = 22/7 * 7 * 7 * 15 cm 3 = 2310 cm 3 Examples Slide 42 Example 2) The area of the base of a right circular cylinder is 154cm 2 and its height is 15cm. Find the volume of the cylinder. Solution) volume of a cylinder = area of the base * height Here, area of the base = 154cm and height= 15cm = (154 * 15) cm 3 Slide 43 Questions 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 cm 3. 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. Slide 44 Base r h l = Slant height Cone Slide 45 3( V ) = r 2 h r hh r Volume of a Cone Here the vertical height and radius of cylinder & cone are same. 3( volume of cone) = volume of cylinder V = 1/3 r 2 h Slide 46 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 Slide 47 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. Slide 48 Example 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= 3m Capacity of the tank = 1/3 r 2 h Capacity of the tank = 1/3 * 22/7 *1.75 * 1.75 * 3 m 3 Capacity of the tank = 9.625 m 3 = 9.625 kilolitre ( 1m 3 = 1 kilolitre) Examples Slide 49 Example 2) The height and the slant height of a cone are 21 cm and 28 cm respectively. Find the volume of the cone. Solution ) l 2 = r 2 + h 2 r = 7 7 cm. volume of the cone = 1/3 r 2 h =1/3 * 22/7 *77 *7 *21 cm 3 = 7546cm 3 Slide 50 Questions 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 314cm 2 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. Slide 51 Surface area 6a 2 2 rh r l4 r 2 Volume a3a3 r 2 h1/3 r 2 h4/3 r 3 Comparison of Area and volume of different geometrical figures Slide 52 V = 4( 1/3r 2 h ) = 4( 1/3r 3 ) h=r r Volume of a Sphere Here the vertical height and radius of cone are same as radius of sphere. volume of Sphere =4( volume of cone) V = 4/3 r 3 r Slide 53 V1 r V=1/3 r 2 h If h = r then V=1/3 r 3 r r 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. V1 = 4V = 4(1/3 r 3 ) = 4/3 r 3 Slide 54 Example 1) Find the volume of a sphere of radius 7cm. Solution) V= 4/3 r 3 Here, r = 7cm V = 4/3 * 22/7 * 7 * 7 * 7 cm 3 = 1437.33 cm 3 Examples. Example 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 r 3 = 2/3 *22/7 * 3.5*3.5*3.5 cm 3 = 89.8 cm 3 Slide 55 Questions 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. Slide 56 Volume of a Cylinder= r 2 h Volume of a Right Circular Cone =1/3 r 2 h Volume of a sphere = 4/3 r 3 Volume of a Hemi sphere = 2/3 r 3 Slide 57 Thank you.