Hence, thickness of the cylinder = me Exercises | zm Take x= 2, unless its value is specified 1. Two closed cuboidal metal tanks are to be painted from inside to avoid rusting. fone tin of paint can paint 5,000 cm? of the area, find which cuboidal tank will require more number Of tins and by how much? 5m 2. Salony is making a twin-colour cuboidal room in her house with di n colour while the base and the lid are white in colour. Find the areas of Walls of the room are blue 2m Tank 1 Ss Fig. 9.32 both the coloured tiles required to make the room. [Picture Based Question| iensions 16 m x6 m x 5m. [Problem Solving Skills] 5. The walls and ceiling of a room are to be plastered. The length, breadth and height of the room are 4.5 m, 3 m and 350 cm respectively. Find the cost of plastering at the rate of 8 per m’, 4. There are two closed boxes as shown in Fig. 9.33. Which box requires the lesser amount of metal sheet to make aid by how much? ‘Gem 50m Fig. 9.33 50cm) [Picture Based Question] 50m oe 5. Aroom has total surface area of 350 m? and lateral surface area of 200 m’, Find the area of its base. 6. If the edge of a cube is doubled, hor any times the surface area will increase? y+ 7 7. Five cubes of side 3 cm each are placed side by side and stuck together to form a cuboid. If the outer surface is to be painted at & 15 per cm’, find the total cost of painting. “ensuration a 163 [Problem Solving Skills)8. Six solid cubical wooden logs with edges 65 cm are to be painted white on all the sides. Fin he quantity of paint required to paint 6 cubical wooden logs if one can of paint is required to pa 11,700 cm? of area. [Problem Solving spin. 9. The ratio between the curved surface area and the total surface area of a right circular cylinder, 1: 2. Find the ratio between the height and radius of the cylinder. 10. Three cubes each of side 10 cm are joined end to end. Find the surface area of the resultant figure 11, Find the area to be painted in the following block Fig. 9.34 with a cylindrical hole. Given that lengy, is 15.cm, width 12 cm, height 20 cm and radius of the hole 2.8 cm. [Picture Based Quests Q 20cm 1am Fig. 9.34 G2) a cuboidal in box ‘opened at the top has dimensions 20 cm x 16 cm x 14 cm. What is the total area of metal sheet required to make 10 such boxes? 15. A cylindrical roller is used to polish a rectangular table. The length of the roller is 0.75 m and the diameter is 0.21 m. If the roller rolls over the table 20 times completely, find the area of the table. 14. The ratio of the curved surface area and the total surface area of a right circular cylinder is 5:7. Find the ratio between the height and radius of the cylinder. [Problem Solving Sk The sum of the area of floor and ceiling of a room is equal to the area of four walls. If the rooms 24m long and 16 m wide. Find the height of the room. 16. In Fig. 9.35, there are two boxes of different shape. Which box has a larger lateral surface area an 10cm (i) / Fig. 9.35 17, Aroller hasa diameter of 0.5 m and a length of 1.5 m. Find the number of revolutions it has to make to cover an area of 264 sqm. [Problen: Solvin, 18. An iron pipe 20 cm long has exterior diameter equal to 25 cm. Ifthe thickness of the pipe find the whole surface area of the pi 1 The total surface area of a hollow cylinder which is open from both sides is 5,740 sq.m, area of be ring is 140 sq. cm and height 10 cm. Find the thickness of the cylinder. 20. A road roller takes 750 complete revolutions to level a road. Find the area of the road if the diame! of the roller is 98 cm and its length 1.5 m. [Problem Solving S' 21. Find the ratio of total surface area and curved surface area of a right citcular cylinder height and radius are 9 m and 2.1 m, respectively. ° 22. The dimensions of a room are 8 m x 6.5 m x 4 m, Find the cost of papering wall paper at the rate of £25 pe [Problem So! —_—_. $$ = 164 Future Kids mathematic’ SS10. n. 12, 13. 14. 15. 16. ———_ 170 bes each of side 0.5 cm is re ao at each edge of a cube is tripled, what will be the change in its volume? the length of : A : den box (including the lid) has external dimensions 40 cm * 34 cm x 30 cm. Ifthe wo, . A woo yaaa quired to build a cube of volume § em’? Jem thick, how many cm’ of wood is used in it? ‘Avwater tank is 1.4 m long, 1 m wide and 0.7 m deep. How many litres of water can it hold? Awe / = 'A cuboid is of dimensions 60 cm x 54 cm x 30 cm. How many small cubes with side 6 cm cany placed in the given cuboid? _ ‘A rectangular sheet of paper is folded about its width 20 cm. Ifthe length of the sheet is 44m, fing the volume of the hollow cylinder thus formed. Ativer 2m deep and 45 m wide is flowing at the rate of 3 km per hour. Find the amount of waterin cubic metres that runs into the sea per minute. Find the number of coins, each of radius 0.75 cm and thickness 0.2 cm, that should be melted to make a right circular cylinder of height 8 cm and base of radius 3 cm. ‘The dimensions of a pool are in the ratio 1 : of the pool. :4. If its volume is 6,144 m’, find the total surface area ‘The inner and outer walls of a 30 cm long pipe have a diameter of 15 cm and 17 cm respectively. Find the volume of the material used in pipe. Find the number of boxes each having a length of 2.4 m, breadth 1.4 m and height 0.8 m that canbe fitted into a room having a length of 6 m, breadth 7 m and height 8 m. The thickness of a hollow metallic cylinder is 2 cm. It is 70 cm long with outer radius of 14 cm. Find) the volume of the metal used in making the cylinder, assuming that it is open at both the ends. Alo find its weight if the metal weighs 8 g per cm’. Radius of a cylinder is r and the height is h. Find the change in the volume if the (i height is doubled. ee (i) height is doubled and the radius is halved. (ii) height remains same and the radius is halved. Russ oe out trough ed Pipe, whose internal diameter is 2 cm, at the rate of 0.7m ye second into a cylindrical tank, the radius of whose base is 40 cm. B i ight © aie cm. By how much will the heigh The dimensions of a metallic cuboid are 100 cm x 80 cm x 64m. Iti i Pe ead the of ana GF thE Gabe cm. It is melted and recast into au A well with diameter 14 m is dug 8 m deep. Thi i al " n leep. The earth taken out of it has b read around it toa width of 21m to form an embankment. Find the height of ane . Future Kids Mather- caangulae plore of paper having length of dal m andl breadth 1.5 mi to be rolled along, ita length oN Naight ciwutar eytinder, Bind the Height and volume of the right circular eylinder, ober \ rhoradins of awell S35 mand the depth of the well below the ground ix 28 m, find the surface area ofthe well, R «Rind the curved surface anea, total surface ana ™ gas is7 em and height being 60 em, and volume of a cylinder, the diameter of whose a Arectangular sheet of paper, 30 em 18 em can be ocular eylinder in two way transformed into the curved surface of right the breadth. Find the rati either by rolling the paper along its length or by rolling it along, 10 of the volumes of the two cy 21, From a tap of inner radius 0.75 em, wat Rtres of water delivered by the linders thus formed, er flows at the rate of 7 pipe in T hour, 2. A square piece of cardboard with sides comer. The remaining flaps are @ volume of the box. m per second. Find the volume in 20 em has a small square of turned up to make a box 4 cm deep. I de 4 cm cut out from each Rind: (8) outer surface area of the box, Give an example to show that the cylinder as comparad to the other cylinder, A rectangular sheet of paper is folded the volume of the hollow tT With greater volume may have lesser lateral surface area about its width 25 cm. If the length of the sheet is 66 cm, find cylinder thus formed, A cylindrical water tank has its diameter 14 m and depth 7 m. How much kilolitres of water can it hold? 25 If the radius of the base of a ri ht circular cylinder is halved, keeping the height same, what is the laced cylinder to that of the original? ~- The diameter of a rolleris 84m and its length is 120m, Ifit takes 500 revolution to le find the area of playground. ratio of the volume of the red vela playground, *S. Find the cost of digging a cuboidal pit 8 m long, 6 m broad and 3 m deep at the rate of € 30 per m?,EXERCISE 9 1. The area of a trapezium shaped field is 3,400 m2 and the length of one of the parallel sides is 100m and its height 40 m. Find the length of other parallel sides. . The area of a trapezium is 686 cm’. If one of the parallel sides exceeds the other by 7. cm and the distance between the parallel sides is 28 cm. Find the length of two parallel sides of the trapezium, 3. Find the area of the following trapeziums: [Picture Based Question] yw (i) Dp 10m Cc (ii) k—_\ 12 mn —— Fig. 9.11 4. The parallel sides of a trapezium are 5 cm and 13 cm and the length of non-parallel sides is 5 om each. Find the area of the trapezium. 5. If one of the parallel sides of a trapezium shaped colony is 260 m, find the other parallel side when the area of the colony is 72,000 m° and the distance between the parallel sides is 180 m. ae 152 Future Kids Mathematics ~§6, The area of a trapezium is 594 sq cm. Its parallel sides are in the ratio 4 : 5. The height of the trapezium is 12 cm. Find the length of the sides. 7. The area of a trapezium is 900 sq m. The distance between the parallel sides is 30 m. If one of the parallel sides is double the other, find the length of the two sides. 8. In Fig, 9.12, there is a book-holder having two trapezium shaped flaps. Find the cost of putting sunmica on the inner side at the rate of & 50 per 10 cm? 40m, — Lew ftw Adem Fig. 9.12 . Mohan wants to buy a trapezium shaped field. Its side along the river is parallel to and twice the side along the road, If the area of the field is 10,500 m? and the perpendicular distance between the two parallel sides is 100 m, find the length of the side along the river. [Problem Solving Skill} 10, In the Fig 9.13, ABCD is a trapezium in which AB||CD. ZBAD = 90°, 9?" —_€ AB = 8 m, CD =3 mand BD = 10 m. Find the area of trapezium. 10m 11. The ratio of the length of parallel sides of a trapezium is 2 : 3. The distance between them is 16 cm. If the area of the trapezium is 320m’, Bm 7B find the length of the parallel sides. Fig. 9.13, 12. Top surface of a dinner table is in the form of a trapezium. Find its area if its parallel sides are 2.5 m. and 1.5 m and perpendicular distance between them is 0.5 m. 13. 14. 15. 16. The diagonals of a’ ‘shombus are 6 cm and 8 cm. Find the length of the side of the rhombus. The floor of a hall consists of 3,000 tiles which are rhombus shaped and ea h of its diagonals are 45 cm and 30 cm in length. Find the total cost of polishing the floor, if the cost perm? is © 4. Find the area of a rhombus, whose side is 5 cm and whose altitude is 4.8 cm, Hone of the diagonals’ is 8 cm long, find the length of other diagonal. [Thinking Skill) Dr. Ritam wants to buy a rhombus shaped plot. If the lengths of diagonals of the plot are 16 m and 30 m, find the area and the length of the side of the field.. SBAENVIVCE 10.4 _ Find the area, in square metres, of the trapezium whose bases and altitudes are as under: (i) bases = 12 dm and 20 dm, altitude = 10 dm Gi) bases = 28 cm and 3 dm, altitude = 25 cm (ii) bases = 8 m and 60 dm, altitude = 40 dm (iv) bases = 150 cm and 30 dm, altitude =9 dm. Find the area of trapezium with base 15 cm and height 8 cm, if the side parallel to the given base is 9 cm long. Find the area of a trapezium whose parallel sides are of length 16 dm and 22 dm and whose height is 12 dm. Find the height of a trapezium, the sum of the lengths of whose bases (parallel sides) is 60 cm and whose area is 600 cm?. Find the altitude of a trapezium whose area is 65 cm? and whose bases are 13 cm and 26cm. Find the sum of the lenghts of the bases of a trapezium whose area is 4.2 m? and whose height is 280 cm. Find the area of a trapezium whose parallel sides of lengths 10 cm and 15 cm are ata distance of 6 cm from each other. Calculate this area as (i) the the sum of the areas of two triangles and one rectangle. (ii) the difference of the area of a rectangle and the sum of the areas of two triangles. The area of a trapezium is 960 cm”. If the parallel sides are 34 cm and 46 cm, find the distance between them.Mathematics for Clags I 18.22 9. Find the area of Fig. 18.35 as the sum of the areas of two trapezium and a rectangle, soon __s0em 1080 4 § oun Fig. 18.35 10. Top surface of a table is trapezium in shape. Find its area if its parallel sides are 1 and 1.2 m and perpendicular distance between them is 0.8 m. a be 1m —+1 fH Fig, 18.36 11. The cross-section of a canal is a trapezium in shape. If the canal is 10 m wide at the top 6 m wide at the bottom and the area of cross-section is 72 m” determine its depth. 12, The area of a trapezium is 91 em? and its height is 7 cm. If one of the parallel sides is longer than the other by 8 cm, find the two parallel sides. 13, The area of a trapezium is 384 cm®, Its parallel sides are in the ratio 3 : 5 and the perpendicular distance between them is 12 em. Find the length of each one of the parallel sides. 14, Mohan wants to buy a trapezium shaped field. Its side along the river is parallel and twice the side along the road. If the area of this field is 10500 m° and the perpendicular distance between the two parallel sides is 100 m, find the length of the side along the river. River Fig. 18.37oe ——$—_Sbezlum and a Polygon)-1 18.23 he area of a trapezium is 158 om. Ione of the parallel sides j ‘phe parallel sides of a trapezium are 25 cm and 13 cm; its nonparallel sides are equal, each being 10 cm, find the area of the trapezium. “ ] Find the area of a trapezium are 15 em each. 6 cm? and the distance between the is 38 cm, find the other. : Parallel sides is 26 16. Whose parallel sides are 25 em, 13 em and the other sides 4g. If the area of a trapezium is 28 em? parallel side if its altitude is 4 em, 19, In Fig. 18.38, a parallelo, 80 cm’, find the area of ¢j and one of its parallel sides is 6 cm, find the other gram is drawn ina trapezium, the area of the parallelogram is he trapezium, Adem 6 em F 4em 120m sem D Baome Fig. 18.38 Fig. 18.39 20. Find the area of the field shown in Fig. 18.39 by dividing it into a square, a rectangle and a trapezium. ANSWERS L @ 16m? (ii) 0.0725 m? Gii) 28m? iv) 2.025m? 2. 96cm? 3. 228m? 1 4 20 em 5. 3 om 6. 3m 7. om 8. 24em 9. 1300cm? 10. 0.88m? 1. 9m 12, 17em,9em 13, 24cm, 400m 4. 140m 15. 84cm 16. 152em® 17. 57/21 em? 18. 8em 19. 128cm? 20. 70cm? 184 AREA OF A POLYGON : In this section, we will discuss some problems on finding the areas of some regular and imegular polygons by using the formulae for the areas of a triangle, rectangle, i i ill be divided into non. ‘um. Infact, given polygon wil = oe relies ace Games .as can be found easily. Thy he Yectilinear plane figures whose are 1 arts, equal to the sum of the areas of non-overlapping par Following examples will illustrate the procedure.N ad _ 10. 1. 12. 13. 14. 15. EXERCISE 19,1 Find the volume of a cuboid whose (@ length = 12 em, breadth =8 em, height = 6 em ra Gi) length = 1.2 m, breadth = 30 cm, height = 15 em (iii) length = 15 cm, breadth = 2.5 dm, height = 8 em, Find the volume of a cube whose side is @4em (ii) Sem (iii) 1.5 dm Find the height of a cuboid of volume 100 cm*, whose length and breadth are (iv) L2m (vy) 25 mm Sem and 4 cm respectively, A cuboidal vessel is 10 em long and 5 em wide. How high must it be made to hold 300 em’ of a liquid? A milk container is 8 cm long and 50 cm wide. hold 4 litres of milk? A cuboidal wooden block contains 36 cm* wood. If it be 4 em long What should be its height so that it can and 3 cm wide, find its height. What will happen to the volume of a cube, if its edge is (i) halved (ii) trebled? . What will happen to the volume of a cuboid if its : G@) Length is doubled, height is same and breadth is halved? (i) Length is doubled, height is doubled and breadth is same? Three cuboids of dimensions 5 em x6 cm x7 em, 4m x7em x8 emand2em x3 cm «13cm are melted and a cube is made. Find the side of cube. Find the weight of solid rectangular iron piece of size 50 cm x 40. em x 10 em, if 1 em? of iron weighs 8 gm. How many wooden cubical blocks of side 25 cm can be cut from a log of wood of size 3 m by 75 cm by 50 cm, assuming that there is no wastage? A cuboidal block of silver is 9 em long, 4 em broad and 3.5 cm in height. From it, beads of volume 1.5 em? each are to be made. Find the number of beads that can be made from the block. Find the number of cuboidal boxes measuring 2 cm by 3 cm by 10 cm which can be stored in a carton whose dimensions are 40 cm, 36 cm and 24 cm. A cuboidal block of solid iron has dimensions 50 cm, 45 cm and 34 cm. How many cuboids of size 5 cm by 3 cm by 2 cm can be obtained from this block? Assume cutting causes no wastage. A cube A has side thrice as long as that of cube B. What is the ratio of the volume of cube A to that of cube B?ration (Volumes and Surtace Ar, cr ors feaS of a Cuboid ay ni Ind a Cube) ream brick measures red in deep fridge AI 16. 0 stot cm inner dimensio - suppose that there are two ety 17. “umes Vand V5 of the cubes MS Are LOO mom many such bricks can be es, having od em by 50 em by 42 em? kes 2 jacket me and compare them," "4 4 em. respectively. Find the toa-ps easures 10 en 4 "< Bom y placed in a cardboard box of dimensions 80 an gtiow Many such tea-packets can be vei S0em = 30cm ~ 02m? The weight ofa metal block of size 5 em by 4em tee ce block of the same metal of size 15 em by Bem by Bac, om | RE: Find the weight of @ ’ ¥ Bem, ay. How many soap cakes can be placed in ofa soap cake is 7m = Sem x 2 cm? 19. a box of size 56 cm x 0.4 m = 0.25 m, if the size a1, The volume of a cuboidal box a ; 3h Tapectively, find its breadth, 4° °™”: Hfits height and length are 3 em and 4 em phages! ANSWERS 1. @ 57cm = 54000 cm? (iii) 3000cm* 2. (i) 64em* (ii) 5IZem* “itd iv) 1728000 cm*=1.728m° —(v) 15.625 em" 3. 5em 4. Gem 5. eae & Som 7 G) § times(ii) 27times 8. (i) same ii) dtimes 9. Sem 10. 160 kg 11. 72 12. 84 13. 576 14, 15. 27:1 16. 150 17. V, =8cm', V, = 64 em’ V, 18. 125 19. 6kg mi 640 21. 4cm 188 OTHER STANDARD UNITS OF VOLUME So far we have used cubic centimetre (cm*) as a standard unit of measurement of volume. Since there are various units of measurement of length like metre, decimetre, decametre ec. Therefore, there are many other standard units of measurement of volume. Also, cubic centimetre is a very small unit for measuring volumes of water tanks, oil tanks ete. We shall now discuss other units of measurements of volume. UTRE OR CUBIC DECIMETRE The volume of the solid region formed by a cube of side I decimetre (dm) is called a litre or a cubic decimetre (1 dm’). 1dm =10em 1dm* =1dmx1dmx1dm =(10%10x10)em* = 1000 em* 7 1 litre = 1000 em* METRE CUBE AND CUBIC METRE The solid region formed by a cube of side 1 m is called a cube and its volume is 1 cubic metre (1 m°). Im =100cm 1m? = (100 x 100 x 100) em® =1000000 em? ut °1000 em® ~ 1 litre 1m? _ 1000000 jitres = 1000 litrestion (Volumes and Surf: ensure face Areas of a Cuboi idand aCube). al 19.15 1, Find the volume in eubie metre ( EXERCISE 19.2 as cu. m) of ea i length = 12m, breadth = 10 m a each of the cuboids whose dimensions are : (ji) length=4 m, breadth =2.5 m, ae (ii) Jength = 10 m, breadth = 25 dm, ia ae ‘i i 4 + 7 =<0 cm. Find the volume in cubie decimetre of each of the cub i @) 15m Gi) 75em Gi) 24m Sem cubes whose side is How much clay is dug out in digi : igging a well measuri ill be th i A uring 3 m by 2m by 5 m? Mane 8 z pe cuboid of volume 168 m’, if the area of its base is 28 m?? he capacity of a ee ons ee 2m high. How much water can it contain? re , “uboidal tank is 50000 lit i \f th tank, fits height and length are 10 m and 2.5 aes Chee Find the breadth of the Arectangular diesel tanker is 2 m long, 2 m wide and 40 1 deep. H diesel can it hold? g and 40 em deep. The length, breadth and height of volume of the air it contains. — ‘A water tank is 3 m long, 2 m broad and 1 m deey hold? How many planks each of which is 3 m long, prepared from a wooden block 6 m long, 75 em broad an‘ th of size 25 em x 10 cm x 8 cm will be required t« it the volume of sand ant flow many litres of m, respectively. Find the p. How many litres of water can it 15 cm broad and 5 cm thick can be d 45 cm thick? . How many bricks eac build a wall 5m Jong, 3 m high and 16 om thick, assuming tha ‘d cement used in the construction is negligible? A village, having a population has a tank which is 20 m long, water of this tank last? 13, A rectangular field is 70 m long and 60m dug outside the field and the earth dug-out much will the earth level rise? M4, Aswimming pool is 250 m long and 130 m wide. 3250 cubic metres of water is pumped into it. Find the rise in the level of water. Jong and 40 em wide contains 0.6 s water per head per day. Tt of 4000, requires 150 litre For how many days will the 15.m broad and 6 m high. broad. A well of dimensions 14m x 8mx 6mis on the field. How from this well is spread evenly cubic metre of wood. How thick is the 15. Abeam 5 m beam? 16, The rainfall on a certain day was 6 om. How many litres of water fell on 8 hectares of field on that day? sliced produces four thousand 1 em cubes am of wood when wood in this process. If one edge of the beam is 0.5 m, find 11. An 8 m long cuboidal be: and there is no wastage of the third edge. 18, The dimensions of a metal block are 2.25 m by 1.5 m by 27 cm, It is melted and recast into cubes, each of the side 45 cm. How many cubes are formed? Ty Asolid rectangular piece of iron measures 6 m by 6 cm by 2cm. Find the weight of this piece, if 1 em? of iron weighs 8 gm.gem is oid of len em. Gv) The volume ofa cube of side ws soem a (w) The volume of wooden cu The height of the ‘cuboid 18 «+ um) nd preadth Gi. g cm is 4000 cm, ii 421.975 4m” Gi) 15.625 dm? ) i) 5m* (i) 625m" 2 ga75dm’ © Teoolitres 8. 675m? a ee c ea 5. 96000 litres 6. 2m ous Ta ae 7 0 11, 1200 yo. gdays so - 0. 9, 6000 litres . Bom Titres . 15, 03 m 16, 18x10 oH ea 7 @ 10° Gi) 1 Gi) 1 Gv) ao! (vii) 10° (viii) 10° Gx) 1 (x) 1000, D PROBLEMS HINTS TO SELECTE! 5. Use: 1m? = 1000 litres 6, Volume of the tank =50000 litres = 50m* 7. Use: 1m* = 1000 litres . 19.9 SURFACE AREAS OF A CUBOID AND A CUBE we have seen thi area of a cuboid equals for the surface area of a cuboid. 1 cm, breadth 6 cm in section 19.2, 30, the surface s now derive the formula ‘onsider a cuboid whose length is ‘ig. 19.9. ‘Area of face ABCD = Area of face EFGH = Area of face AEHD = Area of face BFGC =(bxh) em? Area of face ABEF = Area of face DHGC =(1xh)cm* Total surface area of the cuboid = Sum of the areas of all its six faces =2(Lxb)+2(bxh)+2(Lxh) om? =2(1xb+bxh+lxh) cm? =2(lb+bh+1h) cm? (Lx) em? at the surface of a cuboid co the sum of the areas 0! nsists of six rectangular fa f its six rectangular faces. and height h cm as show A Fig. 19.9 2 (Length x Breadth + Breadth x Height + Length x Height) cm? cm’