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Downloaded from https:// www.studiestoday.com, , VOLUME AND SURFACE AREA, , OF CUBOIDS, , , , ¢ Cubes and Cuboids, ¢ Volume, , Let us first recall the units of volume and their, conversions., , , , Lon*® > = 1 en x1 cm xX lcm, = 10mm x 10 mm x 10 mm = 1000 mm?, ldm? =1dmx1ldmxI1dm, , = 10cm x 10cm X 10 cm, , = 1000 cm* = 1000000 mm? = | @, 1 m? =Ilmxlmxlm, = 100 cm x 100 cm x 100 cm, 1000000 cm’ = 1000 £ = 1 ké, 10m x 10m x 10m = 1000 m’, 1000000 @ = 1000 ké, , 2s Cubes, , Formulae for a cube with length /, > Area of one surface = /? £, > Total surface area of a, cube = 6/7, > Lateral surface area of, cube = 42?, > Volume of a cube =, , 1 dem?, , , , , , , , , , , , ~------ ———_, , , , , , , , , , , , , , Diagonal of a Cube, , A cube has four diagonals. The opposite vertices of, a cube are joined by the diagonals of the cube., In the cube shown in Figure 33.1, the four diagonals, are AG, BH, FD, and EC. Consider diagonal FD. It, does not lie in the same plane as any of the six, surfaces of the cube., , ©, , ¢ Total Surface Area, ¢ Lateral Surface Area, , Consider diagonal DG of square DCGH., DG? = DH? + HG?= # + # = 20, , , , , , , , , , , , => DG = Jj, Now consider rectangle AFGD where, AD =FG= /and A, GD" =AF = 7 L, FD is a diagonal of, rectangle AFGD. |, FD? = DG? + FG? Pee, , ; i, => FD? =(V2e) +2 H*, => FD? = 9/24 2 = 392 Fig. 33.1, , => FD = /3¢, Length of the diagonal of a cube = ./3£, , , , , , , , , , A Cuboids, , Formulae for a cuboid with length /, breadth 4, and, height /, , >» Total surface area of a cuboid = 2(£h + £b + bh), > Lateral surface area of a cuboid = 2h(f + b), > Volume of a cuboid = £x bx h, , , , , , , , , , , , , , , , Downloaded from https:// www.studiestoday.com

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Downloaded from https:// www.studiestoday.com, , Diagonal of a Cuboid, , A cuboid has four diagonals. The opposite vertices, of a cuboid are joined by the diagonals of a cuboid., In the cuboid shown in Figure 33.2, the four, diagonals are AG, BH, CE, and DF., , , , Fig. 33.2, , Consider diagonal EG in rectangle EFGH., EG? = EH? + HG?=2 +, => EG = Jb?+, Now consider diagonal CE in rectangle ACGE., CE? = CG? + EG?, , 2, =n? + (b+ @), , =h? +b? +P, , => CE= JC+P eh, , , , , , , , , , Example 1: If the volume of a cube is 1728 cm’,, find its total surface area and lateral surface area., , Given volume of cube = @° = 1728 cm?, , — = 31728 cm, , = §]2x2x2x2x2K2x3x3x3 cm, a gf2x2x3x2x2x3x2x2x3 cm, , = 2x2x3cm, Thus, the length of the cube is 12 cm., Total surface area of the cube = 6f” = 6 x 12", =6~x 144, = 864 cm?, Lateral surface area of the cube = 40?, = 4% 12?, = 4x 144, = $76 cm’, , , , , , , , Downloaded from https:// www.studiestoday.com, , Example 2: The length, breadth, and height of a, cuboid are in the ratio 7 : 4: 3. If the total surface, area of the cuboid is 5978 cm’, find its volume., , Let the length, breadth, and height of the cuboid, be 7x cm, 4x cm, and 3x cm, respectively., Given total surface area = 2(¢b + bh + Ch), = 5978 cm?, = 2{(7x x 4x) + (4x x 3x) + (7x x 3x)}, = 5978 cm?, => 2(28x + 12x + 21x’) = 5978 cm”, => 2x 61x = 5978 cm? > 122x° = 5978 cm”, 5978, , => ie 2 > J49 =7cm, , Thus, the length = 7 x 7 = 49 cm,, breadth = 4 x 7 = 28 cm,, and height = 3 x 7 = 21 cm, Volume of cuboid = 49 cm X 28 cm X 21 cm, = 28812 cm* = 28.812 dm?, , Example 3: A conference hall is 35.5 m long,, 19.4 m wide, and 6 m high. Its ceiling is covered, with sound absorbing material. Find out how much, it would cost to:, , (i) cover its floor with a wall-to-wall carpet at, Rs 78.50 per sq. m, (ii) paint its walls at Rs 47.50 per sq. m, , Area of floor of hall, = length x breadth = 35.5 m x 19.4m, = 688.7 m?, At Rs 78.50 per sq. m, covering the entire floor with, , carpet would cost, 688.7 x 78.50 = Rs 54062.95, , The lateral surface area of the hall or the area of its, four walls =, Qh(l + b) =2x 6(35.5 + 19.4) m?, = 12 x 54.9 = 658.8 m?, Painting the walls of the hall at Rs 47.50 per m?, , would cost, = 47.50 x 658.8 = Rs 31293.00, , Example 4: A car mechanic, wishing to collect, distilled water, set up a tray | metre in length and, breadth on the roof top and connected a pipe to, drain the tray in a cuboidal tin below that was, , @

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Downloaded from https:// www.studiestoday.com, , 50 cm long, 30 cm wide, and 30 cm high. If 3emof Breadth of the inner cuboid, , rain fell during the day, what was the height of the = 72 cm—3cm-—3cm= 66cm, water that collected in the mechanic’s tin? Similarly as the top and bottom of the box are also, 3 cm thick,, , Rainfall is recorded in terms of height of water, collected, irrespective of how wide or narrow a, container is., , The area of the tray on the roof, , = 100 cm x 100 cm = 10000 cm?, Height of rainwater collected in tray = 3 cm, , , , Thus, volume of rainwater collected, , = 3 x 10000 = 30000 cm? or 30 ¢, When 30 ¢ of water is drained into the empty tin,, volume of water in tin = ¢ X b x h = 30000 cm?, , Fig. 33.3, , height of the inner cuboid, , => 50 cm x 30 cm x f= 30000 cm? = Gite, 8 hi eee eee, aes 3, = hx 1500 cm” = 30000 cm Thus, the volume of the inner cuboid, pl age = 84 cm x 66 cm Xx 54cm = 299376, cm?, 1500 Volume of wood = Volume of outer cuboid, Thus, the height of the water collected in the — Volume of inner cuboid, mechanic’s tin = 20 cm = 388800 — 299376 = 89424 cm*, 3, Example 5: The exterior of an empty wooden box as Usaaisc i ood a hae eae, OX., , measures 90 cm in length, 72 cm in breadth, and, , 60 cm in height. If the wood is 3 cm thick all around,, , find the volume of wood used to make the box. If, , 1 cc of wood weighs 0.09 g find the weight of the, , empty wooden box. Thus, the weight of the empty wooden box is 8 kg, 48.16 g., , Given | cm? of wood weighs 0.09 g,, 89424 cm? of wood weighs, 89424 x 0.09 = 8048.16 ¢g, , Example 6: The length, breadth, and height of a, hall are in the ratio 2: 2: 1. If the lateral surface, area of the hall is 1152 m/’, find the length of its, diagonal., , Lateral surface area = 2h(¢ + 5), Let the length, breadth, and height of the hall be, , 2x, 2x, and x respectively., => (2Xx) (2x + 2x) = 1152 m?*, , , , Volume of the outer cuboid => 2x X 4x = 1152 m?, = 90 cm x 72 cm x 60 cm = 388800 cm? => 8° = 1152 m* ,, If the open box was seen from top it would look like ga, the figure shown in Figure 33.3 As the wood is 3 cm a spent PRT PES: ae, thick on all sides, the length of the inner cuboid Thus, length = 2 x 12 = 24 m, breadth = 24 m, and, = 90 cm —- 3 cm—- 3 cm = 84cm height = 12 m., , ©, , Downloaded from https:// www.studiestoday.com

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Downloaded from https:// www.studiestoday.com, , Length of diagonal = 24 +i Stet, gth of diago C+b+h + Tey t ist |, = [o4? 4024? 412 | 1. Find the volume of a cube. whose, , t [ ; length measures 7 cm,, = [576 +576 +144 ~ 2. Find the volume of a cuboid. given, , its respective length, breadth and, , = 1296 = 36m id, height as 5 cm, 3 cm and Z cm., , Thus, the length of the hall’s diagonal is 36 m., , Exercise 32.3, , 1. Find the total surface area of a cube, given that, , , , 8. Find the length of a cube, given its lateral surface, , , , , , its length is: area is:, (i) 6cm (ii) 7 cm (i) 144 cm? (ii) 484 cm?, (iii) 8.5 cm (iv) 11.4 cm 9. If the volume of a cube is 512 cm’, find its total, , . Find the total surface area of a cuboid, given, , the length, breadth, and height as, (i) 5 cm, 3 cm, and 2 cm, , (ii) 8 cm, 6 cm, and 5 cm, , (iii) 5.5 cm, 3.4 cm, and 2 cm, , (iv) 7.2 cm, 4.5 cm, and 2.4 cm, , . Find the lateral surface area of a cube, given, , surface area and its lateral surface area., , 10. If the volume of a cube is 2744 cm’, find its, , total surface area and its lateral surface area., , 11. Find the measure of the diagonal of a cube,, , given its total surface area is:, (i) 18 cm? (ii) 54 cm”, , 12. Find the measure of the diagonal of a cuboid,, , that its length is: given its respective length, breadth, and height, (i) 4cm (li) 9 cm are:, (ii) 6.5 cm (iv) 10.7 cm (i) 9cm, 3cm, and ji0 cm, 4. Find the lateral surface area of a cuboid, given (ii) 24 cm, 24 cm, and 12 cm, , 7., , 150 cm?, , its respective length, breadth, and height as:, (i) 6 cm, 4 cm, and 2 cm, , (ii) 9 cm, 7 cm, and 5 cm, , (iii) 7.8 cm, 3.2 cm, and 5 cm, , (iv) 9.7 cm, 4.8 cm, and 3.5 cm, , Find the volume of a cube, given that its length 1s:, , (i) 6cm (ii) 9 cm, (iii) 1.1 cm {iv) 3.5 cm, , Find the volume of a cuboid, given its respective, length, breadth, and height as:, (i) 10 cm, 7 cm, and 4 cm, (i) 12 cm, 9 cm, and 3.5 cm, (ii) 8.25 cm, 5.4 cm, and 3.2 cm, (iv) 17.6 cm, 12.2 cm, and 7.5 cm, Find the length of a cube, given its total surface, areais: |, (ii) 486 cm?, , Downloaded from hitps:// www.studiestoday.com, , 13. Aroom is 15.5 m long, 10.8 m wide, and, , 4.5 m high. Find how much it would cost to:, , (i) cover the floor with tiles at Rs 64 per m’,, , (ii) plaster the ceiling at Rs 18 per m’., , (iii) paint the walls at Rs 45 per m?., , , , 14. A hall has 4 windows, each 4 m wide and, , 9 m high, 2 doors, each 2.5 m tall and 2 m wide, , ®

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13,, , 16., , Downloaded from https:// www.studiestoday.com, , and 3 wall closets, each 2.5 m tall and | m wide., If the hall is 20 m long, 18 m wide, and 6 m, high, find the cost of painting its walls at Rs 38, per m’., , The area of the floor of an empty room that is, 4 m high, is 108 m?. Find the volume of air in, , , , the room., , The length, breadth, and height of a cuboid, are in the ratio 6: 5: 3. If the total surface area, of the cuboid is 2016 cm?, find its volume., , 17. The length, breadth, and height of a cuboid are, , 18., , in the ratio 8 : 5: 3. If the lateral surface area, of the cuboid is 702 cm’, find its volume., , An overhead tank 5.6 m long, 2.5 m wide, and, 2 m high is full of water. The entire water is, drained into an empty underground reservoir, 10 m long and 8 m wide. What is the height of, the water in the reservoir now?, , , , 19. A gold bar that measures 20 cm in length,, , 10 cm in breadth and 10 cm in height is melted, , 20., , ai., , 22., , ao, , 24., , 20:, , and recast as gold biscuits 10 cm in length,, 5 cm in breadth, and 2.5 cm thick. How many, gold biscuits were made from the bar of gold?, 8 cm of rainfall was recorded after a, thundershower. How much water fell on a, farmer’s 5 hectare field?, , Rainwater is collected in a tray 1.5 m long and, 1 m wide and then drained into a tin 50 cm, long and 40 cm wide. If the rainfall recorded, was 4 cm, what was the height of the rainwater, that collected in the tin?, , The total surface area of a tank in which the, length, breadth, and height are in the ratio, 2:2: 1 is 5.76 m’. Find the length of the tank’s, diagonal., , The diagonal of an empty cube-shaped tank, , measures 5.19 m. Given 3 = 1.73, how many, , kilolitres of water can be filled in the tank?, , A wooden box is 140 cm long, 100 cm wide,, and 80 cm high. If the wood is 5 cm thick on all, sides of the box, find the volume of wood used, to make the box., , The exterior of an empty steel safe measures, 54 cm in length, 39 cm in width, and 49 cm in, height. If the steel is 2 cm thick all around find:, , (i) the volume of air inside the safe., (ii) the volume of steel used to make the safe., (i) the weight of the empty safe, given that, , 1 cm of the steel weighs 7.5 g., , , , , , surface area and its lateral surface area., Find the measure of the diagonal of a cube, given, its total surface area is:, , (i) 72 cm? (ii) 108 cm?, , . Find the measure of the diagonal of a cuboid given, , its respective length, breadth and height are: 36 cm,, 36 cm and 24 cm., , Revision Exercise, , . If the volume of a cube is 4913 cm’, find its total, , 4, The area of the floor of an empty room that is 6 m, , high, is 114 m2. Find the volume of air in the room., , 6. The length, breadth and height of a room are in, , the ratio 5: 5 : 3. If the lateral surface area of the, room is 7260 m’, find the length of its diagonal., , , , , , ®, , Downloaded from https:// www.studiestoday.com