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For E-books/Materials/Notes-PDFs|PPTs Jobs-Exams-Tests_Papers, etc. More n More...visit: MENSURATION-IV Theory: A solid is a figure bounded by one or more surface. Hence a solid has length, breadth and height. The plane surfaces that bind a solid are called its faces. The fundamental difference between a plane figure and a solid figure is that the plane figure lies in a plane and a solid figure lies in space. There are two types of three-dimensional figures (1) The solid figure in which any of the cross section is the same throughout. E.g. Cube, Cuboid, Cylinder etc. (2) The solid figure in which none of the cross-sections is same throughout. E.g. Cone, Sphere, Pyramid etc. CUBOID: A cuboid is bounded by 6 rectangular faces. The opposite faces of a rectangular solid are equal rectangles lying in parallel planes. E F A B h b G H C D The areas of three different faces be A1 , A2 and A3 then A1 = lb A2 = bh A3 = lh Surface area = 2 ( A1 + A2 + A3 ) = 2 (lb + bh + lh) Volume = Area of any face × corresponding height V = lb × h = lbh Diagonal (d) = l 2 + b 2 + h 2 Diagonal is the biggest possible dimension of a cuboid. Also A1 × A2 × A3 = (lb ) (bh ) (lh ) = (lbh ) 2 = V 2 V= A1 A2 A3 CUBE: a a a A Cube is bounded by six square faces i.e. if the length ,breadth and height of a cuboid are all equal then it is called a cube. If each side of the cube is of ‘a’ units, then its surface area(S.A) =6a2 and Its Volume(V) = a3 Diagonal of cube will be d =

For E-books/Materials/Notes-PDFs|PPTs Jobs-Exams-Tests_Papers, etc. More n More...visit: PROBLEMS 1. Each edge of a cube is decreased by 20%. The percentage of decrease in the surface area of the cube is 1) 44% 2) 36% 3) 20% 4) 60% 5) None of these ANSWER: 2 Edge of the cube be 5 then its surface area = 6 × 52 = 150 4 100 − 20 After reduction new edge of the cube = × 5 = × 5/ = 4 5/ 100 New surface area of the cube = 6 × 42 = 96 150 − 96 54 Surface area reduces by × 100 = × 100 = 36% 150 150 Shortcut method: (i) If each edge of a cube increased by x % then the surface area increases by x2 S = 2 x + % 100 (ii) If each edge of a cube decreased by x % then the surface area decreases by x2 S = 2 x − % 100 20 2 In the above problem x = 20%, then S = 2 × 20 − % = (40 – 4)% = 36% 100 2. A cuboid (3 cm × 4 cm × 5 cm) is cut into unit cubes. The ratio of the total surface area of all the unit cubes to that of the cuboid is 1) 180 : 3 2) 180 : 9 3) 180 : 36 4) 180 : 47 5) None of these ANSWER: 4 The dimensions of a cuboid are 3 × 4 × 5 Its surface area (S.A) = 2(3×4 + 4×5 + 3×5) = 94 cm2 If the cuboid is cut into unit cubes, then the number of unit cubes so formed = 3 ×4×5 = 60 But surface area of each unit cube = 6 × 12 = 6 Total surface area of unit cubes = 6 × 60 = 360 Required ratio = 360 : 94 = 180 : 47 3. If the diagonal of a cube is 10 3 cm, then its surface area will be 2) 550 cm2 3) 600 cm2 4) 650 cm2 1) 500 cm2 these ANSWER: 3 Diagonal (d) of a cube = 10 3 d 10 3 Its side a = = 10 = 3 3 Surface area (S.A) of cube = 6a2 = 6 × 102 = 600 cm2 5) None of

For E-books/Materials/Notes-PDFs|PPTs Jobs-Exams-Tests_Papers, etc. More n More...visit: 4. If the volume of a cube is 216 cm 2 , then the surface area of the cube will be 2) 216 cm 2 3) 218 cm 2 4) 220 cm 2 5) None of 1) 214 cm 2 these ANSWER: Volume (V) of a cube = a3 = 216 a = 3 216 = 6 Its surface area = 6a2 = 6 × 62 = 216 cm2 5. If six cubes, each of 10 cm edge, are joined end to end, then the surface area of the resulting solid will be 1) 3600 cm 2 2) 3000 cm 2 3) 2600 cm 2 4) 2400 cm 2 5) None of these ANSWER: 3 When six cubes are joined end to end, a cuboid will be formed whose length is 6×10 = 60 cm, breadth 10 cm and height 10 cm respectively i.e. l = 60, b = 10 & h = 10 Surface area of cuboid = 2 (60×10 + 10×10 + 60×10) = 2(600 + 100 + 600) = 2600 sq cm 6. If three cubes of copper, each with an edge of 6 cm, 8 cm and 10 cm respectively are melted to form a single cube, then the diagonal of the new cube will be 1) 18 cm 2) 19 cm 3) 19.5 cm 4) 20.8 cm 5) None of these ANSWER: 4 If three cubes are melted to form a single larger cube then the volume of larger cube so formed will be equal to the sum of the volumes of the three cubes. Volume of the larger cube = 63 + 83 + 103 = 216 + 512 + 1000 = 1728 Side of larger cube = 3 1728 = 12 Diagonal of larger cube = 12 3 = 12 × 1.732 = 20.8 cm 7. A swimming pool 9 m wide and 12 m long is 1 m deep on the shallow side and 4 m deep on the deeper side. Its volume is 1) 408 m3 2) 360 m3 3) 270 m3 4) 208 m3 5) None of these ANSWER: 3 The cross-section of the swimming pool is a trapezium whose parallel sides are 1 m and 4 m and having a perpendicular distance of 9 m. 1+ 4 Area of cross-section = × 9 = 22.5 sq m 2 Volume of swimming pool = 22.5 × 12 = 270 cu.m 8. The length, breadth and height of a cuboid are in the ratio 1 : 2 : 3. The length, breadth and height of the cuboid are increased by 100%, 200% and 200% respectively. Then the increase in the volume of the cuboid is 1) 5 times 2) 6 times 3) 12 times 4) 17 times 5) None of these ANSWER: 4 Length, breadth and height of cuboid be x , 2 x and 3 x respectively, then its volume = x × 2 x × 3 x = 6 x 3

For E-books/Materials/Notes-PDFs|PPTs Jobs-Exams-Tests_Papers, etc. More n More...visit: When length, breadth and height are increased by 100%, 200% and 300% respectively, 100 + 100 New length = × x = 2x 100 100 + 200 New breadth = × 2x = 6x 100 100 + 300 New height = × 3x = 9 x 100 New volume = 2 x × 6 x × 9 x = 108 x 3 108 x 3 − 6 x 3 102 x 3 = 17 times Increase in volume = = 6x3 6x3 9. A cube of lead with edges measuring 6 cm each is melted and formed into 27 equal cubes. What will be the length of the edges of the new cubes? 1) 3 cm 2) 4 cm 3) 2 cm 4) 1 cm 5) None of these ANSWER: 3 The edge of each smaller cube be ‘a’. Then total volume of 27 cubes = 27a3 But total volume of 27 cubes is equal to volume of cube of edge 6 cm 27a3 = 63 = 216 216 =8 a3 = 27 a=2 10. The edges of a cuboid are in the ratio 1 : 2 : 3 and its surface area is 88 cm2. The volume of the cuboid is 1) 120 cm3 2) 64 cm3 3) 48 cm3 4) 24 cm3 5) None of these ANSWER: 3 The edges of cuboid are in the ratio of 1 : 2 : 3. So the edges can be assumed as x , 2 x and 3x Surface area (S.A) = 2( x × 2 x + 2 x × 3 x + x × 3 x ) = 2(11x 2 ) = 22 x 2 22 x 2 = 88 x=2 The dimensions of cuboid will be 2, 4 and 6. The volume of cuboid = 2×4×6 = 48 cm3. 11. The areas of three adjacent faces of a cuboid are a,b and c. If the volume of the cuboid is V, then V2 is equal to c 4) (a + b + c) 5) None of 1) abc 2) (ab + bc + ca) 3) ab these ANSWER: 1 If the three adjacent dimensions are x , y and z, then x × y = a y × z = b x × z = c x y = a yz = b x z = c ( x y) (yz) (z x ) = abc

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