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Downloaded from https:// www.studiestoday.com, , , , , , , , , , , , , , , , , , , , Volume, , Definitions and units, , Cuboid, , 1. The volume of a solid figure is the space occupied by it. Volume is measured, in cubic units. The common units of volume and the corresponding units of, length are given in the following table., , Table 2.1 Units of length and volume, , , , , , , , , , , , Unit of length Unit of volume, mm cubic mm (mm?), cm cubic cm (cm?), , m cubic m (m?), , , , , , , , Note The litre (L) is a unit commonly used for measuring the capacity of vessels or the, volume of a liquid., , 1 L = 1 cubic decimetre (dm?) = 1000 cm?, 1 mL = 1cm® orlce, , 2. The surface area of a solid is the sum of the areas of the plane or curved faces, of the solid. It is measured in square units, such as the square centimetre, (em?) and square metre (m7)., , A cuboid is a solid figure bounded by six rectangular, , , , , , , , , , A B, , faces. The adjacent faces are mutually perpendicular and 5 —, , the opposite faces have the same dimensions. A cuboid 1 oe a, , has eight vertices (A, B, C, D, E, F, G, H) and 12 edges ~ t <, , (AB, BC, CD, DA, EF, FG, GH, HE, AH, DE, CF, BG). E F, A cuboid, , The volume of a cuboid is the product of its length,, breadth and height. Denoting the volume, length,, breadth and height by V, I, b and h respectively, we have, , V V V, Vinten t- 5, ka ih lb, , The surface area of a cuboid is the sum of the surface areas of its six rectangular, faces, which works out to the following., , The surface area of a cuboid = 2(lb+ bh+ hl) |, , M-24, , , , , , , , Downloaded from https:// www.studiestoday.com

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Downloaded from https:// www.studiestoday.com, Volume and Surface Area of Cuboids M-25, , The lateral surface area or the area of the four walls of a cuboid works out to., The area of the four walls = 2(l + b)h = perimeter of the floor x height |, , Diagonal of a cuboid, , A diagonal of a cuboid is a line segment joining two, , , , , , , , , , , , , , , , , , vertices which are not on the same face. A cuboid has h ., four diagonals (also called principal diagonals), be, namely HC, AF, BE and DG. All these diagonals are 1 ee, equal in length. Let us find the length of HC. ‘alae, In the rectangle HFCA, HC is the diagonal. eas, HC? = HF? +CF?. aE, , A cuboid has 4 diagonals., , Now, HF is the diagonal of the rectangle EFGH., HF 2 = EF? + EH? =|? +b”., So, HO? =02 £b° 4 CF? S174 b? + h?., , HC =? +b? +h?., , The length of a diagonal of a cuboid = Nie 4b eh? |, , , , EXAMPLE The dimensions of a cuboid are 10 cm by 9.5 cm by 8 cm. Find (i) its volume,, , (ii) its surface area, (iii) the surface area of the four walls, and (iv) the length of a, diagonal., , Solution Here, 1=10cm, b=9.5 cm, h=8 cm., (i) The volume of the cuboid =1x bx h =10x9.5x8 cm? =760cm?., , (ii) Its surface area = 2(Ib+bh+ hl) =2(10x9.5+9.5x8+8 x10) cm?, =2x251 cm? =502 cm?., , (iii) The surface area of the four walls = 2(I + b)h = 2(10 +9.5) x8 em? = 312 cm”., , (iv) The length of a diagonal = Vi? +b? +h? = 4102 +9.52 +87 cm, = /254.25 cm = 15.9 cm (approximately)., , , , , , Cube, , A cube is a solid bounded by six square faces. Its, adjacent faces are perpendicular to each other and all . e, its 12 edges are equal in length. So, a cube is a cuboid 3 f, in which length = breadth = height. a, , The voume of a cube is the cube of the length of its af («a G, , side. Denoting the volume of a cube by V and its length ‘, , b bare: cE, aoe. A cube, , , , , , , , , , , , , , , , , , V=a® and a=%V, , The surface area of a cube is the sum of the areas of its six square faces or, 6 x (length of an edge)”. Denoting the surface area by S,, , S=6a2 and a=2s, , Downloaded from https:// www.studiestoday.com

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Downloaded from https:// www.studiestoday.com, M-26 _ ICSE Mathematics for Class 8, , , , The area of the four walls (lateral surface area) = 4x (length of an edge)? =4a2, , Since a cube is a special cuboid in which | = b = h =a, the length of a diagonal of, a cube = Vl? +b? +h? =V3a? =18a., , The length of a diagonal = /3.a, , EXAMPLE The side of a cube is 6 cm. Find (i) its volume, (ii) its surface area, (iii) its lateral, surface area, and (iv) the length of its diagonal., , Solution (i) The volume of the cube = (length of a side)? =(6 cm)? =216cm?., (ii) Its surface area = 6 x (length of a side)” =6x (6 cm)? =216 cm”., , (ili) The lateral surface area of the cube = 4 x (length of a side)? = 4x (6 cm)?, =144 cm”., , (iv) The length of a diagonal = V3 x length of a side = 6V3 cm., , Solved Examples, , , , EXAMPLE1 The dimensions of a cuboid are in the ratio 4 : 5: 6 and its surface area is, , 5328 m2. Find (i) its length, breadth and height, (ii) its volume and (iii) the length, of a diagonal., , Solution Given that length : breadth : height = 4:5: 6., , Debs an (say) or 1=4xm,b=5xmandh=6xm., , Then the surface area of the cuboid = 2(Ib+ bh + hl), = 2(4x x5x +5x x 6x + 6x x 4x) m?, = 2(20x? +30x? +24x2) m? =148x? m?., , Given, 148x? =5328 or 2 2 0828" a6 OG = 0!, , 148, (i) -. length = 4x6m = 24m, breadth =5x6m = 30m, height = 6 x 6m = 36 m., (ii) The volume of the cuboid =1x bx h = 24x 30x36 m? = 25920 m2., , , , (ili) The length of a diagonal = Vl? +b? +h? = /24? +302 +362 m, = /2772 m=6V77 m., , EXAMPLE 2 The surface area of a cube is 294 cm”. Find (i) the length of an edge of the cube,, (ii) the volume of the cube, and (iii) the length of a diagonal., , Solution (i) Let the length of an edge of the cube = a cm., Then the surface area of the cube = 6a? cm? =294cm? (given)., a - = = 49 or a=7., , So, the length of an edge of the cube = 7 cm., (ii) The volume of the cube =(7 cm)? = 343 cm®., (iii) The length of a diagonal = V3 x a cm = 1.732 x7 cm = 12.124 cm., , Downloaded from https:// www.studiestoday.com

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Solution, , EXAMPLE 4, , Solution, , EXAMPLE 5, , Solution, , Downloaded from https:// www.studiestoday.com, Volume and Surface Area of Cuboids M-27, , The length of a diagonal of a cube is 11/3 em. Find (i) the length of an edge of the, cube, (ii) the volume of the cube, and (iii) the surface area of the cube., , (i) Let the length of each edge = a m., Then the length of a diagonal = /3a cm =11V3 cm (given). “. a=11., So, the length of each edge of the cube = 11 cm., , (ii) The volume of the cube = (11)? cm? = 1331 cm?., , (iii) The surface area of the cube =6x1 12 cm? =726 cm?., , A hall 8 m long, 6 m wide and 4 m high has three doors of size 1.5 m by 2 m and, four windows of size 1.2 m by 1 m. Find the cost of papering the walls if the wall, paper is 80 cm wide and costs ¢ 7 per metre., , Here, =8m,b=6mandh=4m., the area of the walls of the hall = lateral surface area, =~ 2(l+b)h = 2(8 +6)x 4 m? =112 m?., The area of three doors = 3x (1.5 x2) m? =9 m?., , The area of four windows = 4x (1.2 x1) m? = 4.8 m?., the area to be papered = [112 -(9 + 4.8)] m” = (112 -13.8) m2 = 98.2 m”., , The width of the paper = 80 cm = 0.8 m., , the length of paper required = e m= ~ tn =122:75'm., , The cost of papering the walls = % 7 x 122.75 =% 859.25., , The external dimensions of an open wooden box are 26 cm by 24 cm by 15 cm. If, the wood is 2 cm thick, find (i) the internal dimensions of the box, (ii) the capacity, of the box, (iii) the volume of the wood used in making the box, (iv) the cost of, making the box at the rate of = 2/cm®, and (v) the weight of the box if 1 cm® of, wood weighs 0.75 g., , (i) The internal dimensions of the box are as follows., The internal length = 26 cm—2 x2 cm=22 cm., The internal breadth = 24 cm-2 x2 cm= 20 cm., , The internal height = 15 cm-2 cm=13 cm BA ay, (.:. the box is open at the top). 26 cm, , 2cm, , , , 15cm, , , , , , , , , , (ii) The capacity of the box = the internal volume of the box, = 22 x20x13 cm® =5720 cm’., (iii) The volume of wood = the external volume of the box - the internal volume, of the box, = (26 x 24x 15 -5720) cm®, ~ (9360 -5720) cm® = 3640 cm’., , (iv) The cost of making the box = ¥ 2 x 3640 = = 7280., , (v) The weight of the box = the weight of the wood, = 3640 0.75 g = 2730 g =2 kg 730 g., , EXAMPLE6 The dimensions of a cuboid are equal to half the dimensions of another cuboid., , Find the ratio of the (i) volumes, (ii) surface areas of the two cuboids., , Downloaded from https:// www.studiestoday.com

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Downloaded from https:// www.studiestoday.com, M-28 ICSE Mathematics for Class 8, , Solution Let the dimensions of the bigger cuboid be length = I, breadth = b, height = h., Then those of the smaller cuboid are length = st breadth = 5b height = st, , (i) The volume of the bigger cuboid = lbh., , The volume of the smaller cuboid = =! x 5b x sh = a lbh., , 1, , volume of the smaller cuboid g en, , ~ =—=1:8., volume of the bigger cuboid JIbh 8, , , , (ii) The surface area of the bigger cuboid = 2(lb + bh + hl)., , The surface area of the smaller cuboid = a5! x Lup — x ip Sone 31), 2.822, 2, 2 2 2}, = 2(Ib + bh+hl) x2, , 2b +bh+hl) x7, , surface area of the smaller cuboid _ x i Lay PA, , surface area of the bigger cuboid = 2(Ib + bh + All) 4, , , , Remember These, , , , 1. For a cuboid of length |, breadth b and height h:, , (i) Volume (V)—Ixb<f [-— : b= ee b=, x, , bxh’ ixh, (ii) Surface area (S) = 2(lb + bh + hl), (iii) Surface area of the four walls (lateral surface area) = 2(1+b)h, (iv) Length of a diagonal = VI? +b? +h?, , 2. For a cube of edge a:, , , , , , , , (i) Volume (V)=a?, a=3/V (ii) Surface area (S)=6a?, a= -s, , (iii) Surface area of the four walls = 4a? (iv) Length of a diagonal = /3a, , , , , , 1. Find the volume, surface area, lateral surface area and length of a diagonal of a cuboid with, dimensions:, , (i) 8cm by6cm by5cm (ii) L2emby9cmby8cm (iii) 16cm by 12 cmby 10cm, , 2. Find the volume, surface area, lateral surface area and length of a diagonal of a cube, of edge:, , (i) 7 cm (ii) 10 cm (iii) 4 cm, , Downloaded from https:// www.studiestoday.com