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Downloaded from https:// www.studiestoday.com, , CHAPTER 33, , VOLUME AND SURFACE AREA, (Cuboid and Cube), , , , , , 33.1) INTRODUCTION, , , , , , , , , , , , , , , , , , , , Volume The space occupied by a body (solid) is called its volume., , Surface area | The sum of areas of all the faces of a body is called its surface area., , Units of length Unit of volume Unit of surface-area |, m (metre) m® (cubic metre) m2 (square metre) i, cm cm? cm? ;, , mm mm? mm?, , Also,, , 1, 1 m3 = 100 x 100 x 100 cm = 1000000 cm$ and 1 cm* = 799x499x400, 1, 1 cm? = 10 x 10 x 10 mm? = 1000 mm and 1 mm = 7O9q cm?, , In general, the volume of a liquid or a gas is measured in litres, such that, 1 m? = 1000 litre and 1 litre = 1000 cm$ (c.c. or millilitre), , , , , , , , , , 33.2 | CUBOID (a rectangular solid), , , , , , A cuboid is a solid bounded by six rectangular faces., , 1. Volume of a cuboid, = its length x breadth x height 7, = xin A, , , , , , a Total surface area of a cuboid = Area of six rectangular faces, Since, Area of ABCD + Area of EFGH = 2(/ x b) [Opposite faces are equal], Area of BCGF + Area of ADHE = 2(bx h) [Opposite faces are equal], and Area of ABFE + Area of DCGH = 2(h~x J) [Opposite faces are equal], , Total surface area of cuboid = 2(/ x b+ bx h+hx/l), , , , , , , , 33.3} CUBE, A cube is a rectangular solid whose each face is a square., In other words, a cube is a cuboid whose, length = breadth = height = a —, 1. Since volume of a cuboid =/xbxh, Volume of a cube = axaxa, = a? = (its edge)?, 2. Total surface area of a cube = 2(axa+axa+axa), , , , , , , , = 6a? = 6 (edge)? a a, a, Example 1 :, The length, breadth and height of a cuboid are in the ratio 6 : 5: 4. If its volume is, 15,000 cm’; find : (i) its dimensions (ii) its surface area., ce oe Downloaded from http84vww.studiestoday.com ef a

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Downloaded from https:// www.studiestoday.com, , , , , , (i) Given: Length : breadth : height = 6:5: 4, = If length = 6x cm, breadth = 5x cm and height = 4x cm, Length x breadth x height = volume, , , , = 6x x 5x x 4x = 15,000, => x3 = eee =125=5x5x5=5°9, 6x5x4, x=5, , i.e. length = 6x cm = 6 x 5 cm = 30 cm, breadth = 5x cm = 5 x 5cm = 25 cm, and, height = 4x cm = 4 x 5cm = 20 cm (Ans.), (ii) Surface area of the cuboid = 2(1x b+ bx h+hx]), = 2(30 x 25 + 25 x 20 + 20 x 30) cm?, = 2(750 + 500 + 600) cm? = 3700 cm? (Ans.), , Example 2:, The total surface area of a cube is 294 cm/, find its volume., , Solution :, , Since total surface area of cube = 6 x (side)?, , => 6 x (side)? = 294, , = side = 7 cm, , “ volume = (side)? = (7 cm)? = 343 cm? (Ans.), Example 3, , A rectangular solid of metal has dimensions 50 cm, 64 cm and 72 cm. It is melted and, recast into identical cubes each with edge 4 cm, find the number of cubes formed., , Solution :, .. Volume of rectangular solid melted = its length x breadth x height, = 50 x 64 x 72 cm?, And, volume of each cube formed = (its edge)?, = (4)3 cm’ = 4 x 4 x 4. cm, Volume of solid melted, , , , , , , , , , , , , , , , , , , , , , , , , , , Number of cubes formed = ~V/7i ime of each cube, 50x64x72 _, Twa a OF 3600 (Ans.), Example 4:, , Three cubes, each of edge 8 cm, are joined as E, shown alongside. Find the total surface area and the «©, volume of the cuboid. ;, , §, aerate“ san ®, , wnloaded from https:// www.studiestoday.com

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Solution :, , um ell tse : |, , , , Downloaded from https:// www.studiestoday.com, , length (/) of the resulting cuboid = 3 x 8cm=24cm,, , its breadth (b) = 8 cm and its height (h), 2(ixb+bxh+hx l), 2(24 x 8+ 8x8 +8 x 24) cm? = 896 cm?, Ixbxh, = 24x 8x8 cm = 1536 cm?, , Since,, =8cm, , Total surface area, , Volume, , , , , , , 41m ta om, mie = ae Nii fra, a OMe na cm? and, A= oe % ses coniens seas Gm Ss aac: Se *cm$, , Bf a1 e.g litre and’ lithe = ...s.-accc.- cm3,, , 3. A cube is always a.............0 ; but a cuboid is not necessarily a ....... sae, , 4. The volume of a cube with side a cm is numerically equal to its surface area: then, Teor sesso CES ceen rer orereeen anda = see, , 5. Each edge of a cube is 8 cm; area of each face of the cube = ............. Meee a, Se cnt cm? and total surface area of the cube is ................. alee ite, , 6. Each edge of a cube is doubled, then its total surface area becomes ...............+. times |, and its volume becomes ............1++ times. ie, , 7. Asolid cuboid (36 cm x 3 cm x x cm) has the same volume as a solid cube of edge >, 6 cm; then.2scese oe, Se cca teene ees ee ANG IMs on sees ales s)s PE gomee nee, , 8. A cubical container, with each edge 10 cm; is full of water. This water is transferred to, an empty rectangular container with length 20 cm and breadth 5 cm. If the height of, water in the rectangular container is x cm, then 10 x 10 x 10 = ......ceeeeseeeeees Seanadeasfiade, GANG Xi ea cocenpas sncewrap esac gn deccateeseest cesucasuseeeceseras, , , , ile, , as, , , , EXERCISE 33 (A), , in the ratio 6 : 5: 3. If its total surface area is, 504 cm2, find its dimensions. Also, find the, volume of the cuboid., , Find the volume and total surface area of a, cube whose each edge is :, , (i) 8cm (ii) 2m 40cm., , Find the volume and the total surface area of, a cuboid, whose :, , (i) length = 15 cm, breadth = 10 cm and, height = 8 cm 5., , (ii) 1=3-5m,b=2-6 mandh=90 cm., (i) The volume of a cuboid is 3456 cm‘%. If, , ee, , its length = 24 cm and breadth = 18 cm, 6., , find its height., , (ii) The volume of a cuboid is 7-68 m‘°. If its, length = 3-2 m and height = 1-0 m; find, its breadth., , (iii) +The breadth and height of a rectangular, solid are 1-20 m and 80 cm respectively., , If the volume of the cuboid is 1-92 m3,, find its length., , The length, the breadth and the height of a, cuboid are in the ratio 5 : 3: 2. If its volume is, 240 cm’; find its dimensions. Also, find the, total surface area of the cuboid., , The length, breadth and height of a cuboid are, , , , Find the length of each edge of a cube, if its, volume is :, , (i) 216 cm? (ii) 1-728 m8, The total surface area of a Cube is 216 cm?., Find its volume., , A solid cuboid of metal has dimensions 24 cm,, 18 cm and 4 cm. Find its volume., , A wall 9 m long, 6 m high and 20 cm thick, is, to be constructed, using bricks of dimensions, 30 cm, 15 cm and 10 cm. How many bricks, will be required ?

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Downloaded from https:// www.studiestoday.com, , , , , , , , , , , , , , , , , , 10. A solid cube of edge 14 cm is melted down 12. Four cubes, each of edge 9 cm, are joined as, and recasted into smaller and equal cubes shown below :, each of edge 2 cm, find the number of smaller ; : : A, cubes obtained. ; tg om, 11. A closed box is cuboid in shape with iPecllie eee eel omer real eUme sw Lo. .c. v, length = 40 cm, breadth = 30 cm and Pelee ac |e “in, height = 50 cm. It is made of thin metal eee eee, sheet. Find the cost of metal sheet required 9cm 9cm <9cm = 9em, to make 20 such boxes, if 1 m? of metal sheet Write the dimensions of the resulting cuboid, costs & 45. obtained. Also, find the total surface area and, the volume of the resulting cuboid., , , , , , a, , [33.4] APPLICATION, , 1. Foraroom:, Every room has four walls; two walls along its length and two walls along its width., (i) Area of each wall along the length = 1x h, and, (ii) Area of each wall along the width = bx h, ‘ Area of 4 walls of the room = 2x/xh+2xbxh, = 2(l1+b)xh, , , , ‘the area of doors and windows., , , , Also, (iii) The area of roof = the area of floor =/ x b, , Example 5 :, , The internal length, breadth and height of a rectangular room are 6 m, 5-2 m and 4-5 m, respectively. It has two doors each of 1-2 m by 2 m and three windows each of 1 m by 80 cm., Find the total internal area of the room to be whitewashed., , ;, |, |, :, :, , Also, find the cost of whitewashing the room (excluding the doors and windows) at the, , rate of = 6 per m?., | Solution :, For the room, its /= 6m, b=5-2mandh=45m, | .. Area of its four walls = 2(/+ b)h, = 2(6 + 5-2) x 4-5 m2 = 100-8 m?, , Area of its roof = 1x b=6 x 5-2 m? = 31-2 m2, , Since, area of one door = 1-2 x 2 m2 = 2-4 m?, * area of two doors = 2x 2-4 m? = 4-8 m2, Also, area of each window = 1 x 0-80 m2 =0-:80 m2 [80 cm = 0-80 m], , Area of three windows = 3 x 0-80 m? = 2-40 m2, Total internal area of the room to be whitewashed, , = (Area of four walls + Area of roof) — (Area, of two doors + Area of three windows), , = (100-8 + 31-2) — (4-8 + 2-4) m2, = 124.8 m2 (Ans.), Cost of whitewashing = % 6 x 124-8 = = 748-80 (Ans.), , EE 327 ~~ zl, , Downloaded from https:// www.studiestoday.com

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Downloaded from https:// www.studiestoday.com, , 2. For a box:, (i) Space occupied by it = its external volume, (ii) Its capacity = its internal volume, , (iii) Volume of material in it = its external volume — its internal volume., , 3. For a closed box :, , If its external length, breadth and height are /, b and h respectively, and its walls are x, unit thick throughout, then :, , (i) Its internal length = External length — twice the thickness of walls, = 1-2x, (ii) Its internal breadth = b-2x and, (iii) Its internal height = h — 2x, , Conversely, if the internal dimensions of a box are /, b and h respectively and its sides, (walls) are x unit thick everywhere, then its external dimensions are / + 2x, b + 2x and h + 2x, respectively., , Example 6 :, , The external length, breadth and height of a closed wooden box are 30 cm, 18 cm and, 20 cm respectively. If the walls of the box are 1-5 cm thick, find :, , (i) capacity of the box;, (ii) volume of the wood used in making the box;, and (iii) weight of the box; if 1 cm? of the wood weighs 0-80 g., , Solution :, Given, external length of the box = 30 cm, external breadth of the box = 18 cm, and, external height of the box = 20 cm, External volume of the box = 30 x 18 x 20 cm®, = 10,800 cm?, , Since, the walls of the box are 1-5 cm thick throughout;, Internal length of the box = (30-2 x 1-5) cm = 27 cm, internal breadth of the box = (18-2 x 1-5) cm=15cm, , and, internal height of the box = (20 —2 x 1-5) cm=17 cm, Internal volume of the box = 27 x 15 x 17 cm%;, = 6,885 cm? e, (i) Capacity of the box = its internal volume, = 6,885 cm? (Ans.), (ii) Volume of the wood used = External volume — Internal volume, = 10,800 cm? — 6,885 cm?, = 3,915 cm? (Ans.), (iii) Since, 1 cm® of wood weighs 0-80 g, ; Weight of the box = 3,915 x 0-80 g, = 3132 g = 3-132 kg (Ans.), , ee Downtoaded from http SPR studiestodey Come