5) surface areas & volumes - questions

1. One side of a right triangle measures 126 m and thedifference in lengths of its hypotenuse and other side is42 m. Find the measures of its two unknown sides andcalculate its area. Verify the result using Heron’sFormula

[Hint : C - b = 42 Þ c = 42 + b & c2 = a2 + b2Þ (42 + b)2 = (126)2 + b2]2. Using Heron’s Formula, find the area of an equilateraltriangle the length of one side is a.

3. Find the area of an isosceles triangle, the measure ofone of its equal sides being b and the third side a.

4. Find the area of a right angled triangle if the radius of itscircumcircle is 3 m and altitude drawn to thehypotenuse is 2 cm.

[Hint :Let ABC be the right angled triangle right angled at B. LetO be the centre of the circumcircle. Them by geometryO is the mid-point of the hypotenuse AC.]

5. A regular hexagon has a side 8 cm. Determine itsperimeter and area.

[Hint :Area of hexagon = 6 x area of equilateral triangle OAB]

6. The perimeter of right triangle is 90 cm. Its hypotenuseis 41 cm. Find the other two sides and the area of thetriangle.

[Hint :a + b + 41 = 90 Þ a + b = 49 cmAlso, a2 + b2 = (41)2 or (49 - b)2 + b2 = (41)2]

7. An isosceles right triangle has an area 200 cm2. What isthe length of its hypotenuse?

[Hint :

8. Radha made a picture of an aeroplane with colouredpaper as shown in figure, Find the total area of thepaper used.

9. A triangle and a parallelogram have the same base andthe same area. If the sides of the triangle are 26 cm,28cm and 30 cm, and the parallelogram stands on thebase 28 cm, find the height of the parallelogram.

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10. In figure, OPQR is a rhombus, three of whose verticleslie on the circle with centre O. If the area of the

rhombus is , find the radius of the circle.[Hint :

Area of equitateral DOQR + Area of equitateral

DOPQ =11. A parallelogram, the measures of whose sides are 25 cm

and 15 cm has one diagonal 20 cm long. Find its altitudeon the side 25 cm.

12. The base of a triangular field is three times its height. Ifthe cost of cultivating the field at Rs. 300 per m2 is Rs.181250 , find its base and height. [NCERT][Hint : If height = h m, then base = 3h m. Area of field =

Rs. ]13. Find the square of the radius of te circle whose area is

the sum of the area of two triangles whose sides are 35,53, 66 and 33, 56, 65 measured in centimeters. (Take p= 22/7)

14. Find the area of a quadrilateral field whose diagonalsmeasure 48 m and 32 m and intersect each other atright angles. Find also the cost of the land at the rate ofRs/ 7000 per square metre.[Hint :

Required area

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15. A trapezium PBCQ, with parallel sides QC and PB in theratio of 7 : 5, is cut off from a rectangle ABCD as shown

in the following figure. If the area of the trapezium ispart of the area of the rectangle. find the lengths of QCand PB.

16. The perimeter of an equilateral triangle measuretimes metres as the area of the equilateral trianglemeasures square metres. Find the length of its side.[ Hint : Let the length of the side be ´ m. Then 3x =

]

17. In an equilateral triangle of side 2a units, find the lengthof its altitude.


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18. A like in the shape of a square with a diagonal 40 cm andan idosceles triangle of base 10 cm and sides 13 cm eachis to be made of three different shades as shown in fig.How much paper of each shade has been used in makingthe kite?

19. In figure, AB = 28 m, AC = 24 m, BC = 20 m, CG = 32 m,AG = 40 m and D is mid-point of AG. Find the area of thequadrilateral ABCD.

20. White and grey coloured triangular plastic sheets areused to make a toy as shown in fig. Find the total areasof white and grey coloured sheets for making the toy.

21. Suman made an arrangement with white and blackcoloured paper sheets as showing in fig. Find the totalareas of the white and black paper sheets used inmaking the arrangement.

22. A floral design on a floor is made up of 16 tiles which aretriangular, the sides of the triangular tiles are 26 cm, 20cm, and 10 cm. The tiles are polished at the rate of 20p per cm2. Find the cost of polishing the tiles. (Take

= 3.74)


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23. Suman made a picture with some white paper and asingle coloured paper as showing in fig. White paper is aavailable at her home and free of cost. The cost ofcoloured paper used is at the rate of 10 p per cm2. Find

the total cost of the coloured paper used. (Take =

1.732 and = 3.31)

24. In figure, P and Q are two lamp posts. If the area of theDPBC is same as that of the rectangle ABCD, find thedistance between the two lamp posts.

25. A triangle and a parallelogram has same base and samearea. If the sides of the triangle are 20 cm, 25 cm and 35cm, and the base side is 25 cm for the triangle as well asthe parallelogram, find the vertical height of theparallelogram.

26. A triangle and a parallelogram have a common side andare of equal areas. The triangle having sides 26 cm, 28cm and 30 cm stands on the parallelogram. Thecommon side of the triangle and the parallelogram is 28cm. Find the vertical height of the triangle and that ofthe parallelogram.

27. A farmer has two triangular fields in the form of DABCand DACD in which the side AC is common as shown infigure. AB = 840 m, BC = 600 m, AC = 480 m, AD = 800m, AD = 800 m and CD = 640 m. He has marked midpoints E and F on the sides AB and AD respectively. Byjoining CE and CF, he has made a field in the shape ofquadrilateral AECF. He grew wheat in the quadrilateralplot AECF, potatoes in DCFD and onions in DBEC. How

much are has been used for each crop ? (Take =2.45 ; one hectare = 10000 m2).

28. A field in the form of quadrilateral ABCD whose sidestaken in order are respectively equal to 192, 576, 288and 480 dm has the diagonal equal to 672 dm. Find itsarea to the nearest square metre.

29. A trapezium with its parallel sides in the ratio 16 : 15 iscut from a rectangle whose sides measure 63 m and 5 m

respectively. The area of the trapezium is of thearea of the rectangle. Find the lengths of the parallelsides of the trapezium.

30. Find the cost, at Rs. 25 per 10 square metres, of turfing aplot of land in the form of parallelogram whose adjacentsides and one of the diagonals measure 39 m, 25 m and56 m respectively.

31. A triangular park in a city has dimensions 100 m × 90 m× 110 m. A contract is given to a company for plantinggrass in the park at the rate of Rs. 4000 per hectare. Find

the amount to be paid to the company. (Take =1.414) (one hectare = 10,000 m2)


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32. There is a slide in a children park. The front side of theslide has ben painted and a message “ONLY FORCHILDREN” is written on it as shown in fig. If the sides ofthe triangular front wall of the slide are 9 m. 8 m and 3m, then find the area which is painte in colour.

33. The perimeter of a triangular park is 180 m and its sidesare in the ratio 5 : 6 : 7. Find the area of the park.

34. A triangle has sides 35 mm, 54 mm and 61 mm long.What is its area. Find also the smallest altitude of thetriangle.

35. The perimeter of a right triangle is 12 cm and itshypotenuse is of length 5 cm. Find the other two sidesand calculate its area. Verify the result using Heron’sFormula.

36. Using Heron’s Formula, find the area of an isoscelestriangle, the measure of one of its equal sides being aunits and the third side 2b units.

37. The sides of triangle are 39 cm, 42 cm, and 45 c m. Aparallelogram stands on the greatest sides of thetriangle and has the same area as that the triangle. Findthe height of the parallelogram.

38. From a point in the interior of an equilateral triangleperpendiculars drawn to the three sides are 8 cm, 10 cmand 11 cm respectively. Find the area of the triangle to

the nearest cm. (use )

39. A municipal corporation wall on road side hasdimensions as shown in fig. The wall is to be used foradvertisements and it yields an earning or Rs. 400 perm2 in a year. Find the total amount of revenue earned ina year.

40. ABCD is quadrilateral such that AB = 5 cm, BC = 4 cm, CD= 7 cm, AD = 6 cm and diagonal BD = 5 cm. prove that

the area of the quadrilateral ABCD is cm2.41. Find the area of the quadrilateral ABCD in which AB = 7

cm, BC = 6 cm, CD = 12 cm, DA = 15 cm and AC = 9 cm.

(Take = 10.5 approx.)42. A rhombus has perimeter 64 m and one of the diagonals

is 22 m. Prove that the area of the rhombus is

43. ABCD is a trapezium in which AB }} CD ; BC and AD arenon-parallel sides. It is given that AB = 75 cm, BC = 42cm, CD = 30 and AD = 39 cm. Find the area of thetrapezium.

44. OABC is a rhombus whose three vertices A, B and C lieon a circle with centre O. If the radius of the circle is 10cm. find the area of the rhombus.


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45. The cross-section of a canal is in the shape of atrapezium. If the canal is 12 m wide at the top and 8 mwide at the bottom and the area of its cross-section is84 m2, determine its depth.

46. Students of a school staged a rally for cleanlinesscampaigns. They walked through the lanes in twogroups. One group walked through the lanes AB, BC andCA ; while the other through AC, CD and DA. Then theycleaned the area enclosed within their lanes. If AB = 9m, BC = 40 m. CD = 15 m, DA = 28 m and ÐB = 900,which group cleaned more area and by how much ? Findthe total area cleaned by the students.

47. Find the perimeter of a square, the sum of lengths ofwhose diagonals is 144 cm.

48. Find the area of a quadrilateral piece of ground one ofwhose diagonals is 60 metres long and theperpendiculars from the other two vertices are 38 and22 metres respectively.

49. Write the area of a triangle having 5 cm base and height6 cm.

50. Write the area of an equilateral triangle whose side is 6cm.

51. State Heron’s Formula for area of a triangle.52. In DABC, BC = a, CA = b and AB = c. Write the

semiperimeter s.53. Find the area of isosceles triangle ABC in which AB = AC

= 5 cm and BC = 8 cm.

54. Find the area of isosceles triangle having each side oflength a cm.

55. Find the area of the triangle having three sides given as5 cm, 6 m and 7 cm.

56. Three equal cubes are placed adjacently in a row. Findthe ratio of the total surface area of the new cuboid tothat of the sum of the surface areas of three cubes.[Hint :

Let the side of a cube be a units. Then, for the resultingcuboid, we have,

Length = a + a + a = 3a units.Breadth (b) = a units.Height (h) = a units.]

57. The floor of a rectangular hall has a perimeter 250 m. Ifthe cost of painting the four walls at the rate of Rs. 10per m2 is Rs. 15000, find the height of the hall.[Hint : Let the height of the hall be h m.Area of 4 walls = 2( + b) h = perimeter × hThen, 250 × h × 10 = 15000]

58. Ajay has built a cubical water tank in his house. The topof the water tank is covered with lid. He wants to coverthe inner surface of the tank including the lid withsquare tiles of side 25 cm. If each inner edge of thewater tank is 2 m long and the tiles cost Rs. 360 perdozen, then find the total amount required for tiles.[Hint : No. of tiles required =

Total cost = Rs. × No. of tiles ]59. The length of the hall is 20 m breadth 16 m. The sum of

the areas of the floor and the flat roof is equal to thesum of the areas of the four walls. Find the height of thehall.[Hint : ( × b) + ( × b) = 2 ( + b) h]


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60. Shanti Sweets Stall was placing an order for makingcardboard boxes for packing their sweets. Two sizes ofboxes were required. The bigger of dimensions 25 cm ×20 cm × 5 cm and the smaller of dimensions 15 cm × 12cm 5 cm. For all the overlaps, 5% of the total surfacearea is required extra. If the cost of the cardboard is Rs.4 for 1000 cm2, find the cost of cardboard required forsupplying 250 boxes of each kind.

61. The sum of the length, breadth and height of a cuboid is21 cm and the length of its diagonal is 13 cm. Find thesurface area of the cuboid. Also find the cost of paintingthe surface at the rate of Rs. 1.40 per cm2.

[Hind : + n + h = 21 & = 13 or +b2 + h2 = 169

( + b + h)2 = + b2 + h2 + 2( + bh + )

Þ 2( + bh + ) = ( + b + h)2 - ( + b2 + h2)]62. The cost of papering the four walls of a room at 70 paise

per square metre is Rs. 157.50. The height of the room is5 metres. Find the length and the breadth of the room ifthey are in the ratio 4 : 1.[Hint : Let length be 4x m and breadth be x m.

2(4x + x) (5) × = 157.50]63. The capacity of a cuboidal tank is 50000 litres of water.

Find the breadth of the tank, if its length and depth arerespectively 2.5 m and 10 m.[Hint : 50,000 litres = 50 m3]

64. A rectangular water reservoir is 10.8 m by 3.75 m at thebase. Water flows into it at the rate of 18 m per secondthrough a pipe having cross section 7.5 cm × 4.5 cm.Find the height to which the level of water reach in 15minutes.

65. A solid cube of side 12 cm is cut into eight cubes ofequal volume. What will be the side of the new cube ?Also, find the ratio between their surface areas.[Hint : Let the side of new cube be a cm.Volume of bigger cube = 8 × volume of a smaller cube]

66. The areas of three adjacent faces of a cuboid are p,q andr. If its volume is v, prove that v2 = pqr.[Hint

p = b × h ; q = × b ; r = xh Þ pqr = ]67. In figure, you see the frame of a lampshade. It is to be

covered with a decorative cloth. The frame has a basediameter of 20 cm and height of 30 cm. A margin of 2.5cm is to be given for folding it over the top and bottomof the frame. Find how much cloth is required forcovering the lampshade.

[Hint : r = 10 cm, h = (30 + 2.5 + 2.5) cm = 35 cm]68. If the radius of the base of a right circular cylinder is

halved, keeping the height same , what is the ratio ofthe volume of the reduced cylinder to that of theoriginal one ?[Hint : let r be the radius, h be the height & v1, be thevolume of the original cylinder. The, for

reduced cylinder, we have radius = , height = h andvolume = v2]

69. If costs Rs. 2200 to paint the inner curved surface of acylindrical vessel 10 m depp. If the cost of painting is atthe rate of Rs. 20 per m2, find(i) Inner curved surface area of the vessel,(ii) Radius of the base,(iii) Capacity of the vessel.[Hint : Inner curved surface area × Rs. 20 = Rs. 2200]


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70. Water is supplied to a city population for general use(not for drinking) from a river through a cylindrical pipe.The radius of the cross-section of the pipe is 20 cm. Thespeed of water through the pipe is 18 km per hour. Findthe quantity of water in litres which is supplied to thecity in two hours. (Take = 3.14 and 1 m3 = 1000litres.)[Hint : 18 km/hr = 18000 m /hrdistance covered by water in 2 h = 36000 m

Radius of cross section = 20 cm = m

Volume of water which flows in two hours = 3.14 ×× 18000 × 2 m3

= × 1000 litres]71. The capacity of a closed cylindrical vessel of height 1 mis 15.4 litres. How many square metres of metal sheetwould be needed to make it ?[Hint : Let the radius of the vessel be r m.

Volume of vessel = 15.4 = 0.0154 m3 Þ × r2× h = 0.0154]

72. A corn cob, shaped like a right circular cone, has theradius of its broadest end as 2.1 cm and length (height)as 20 cm. If each 1 cm2 of the surface of the cob carriesan average of four grains, find the number of grains onthe entire cob.

[Hint: Number of grains on the entire cob = 4 × curvedsurface area.]73. What length of tarpaulin 3 m wide will be required to

make conical tent of height 8 m and base radius 6 m?assume that the extra length of material that will berequired for stitching margins and wastage in cutting isapproximately 20 cm (Use = 3.14)[Hind : Area of Tarpaulin required = Curved surface ofthe conical tenti.e., × b = ]

74. A right triangle ABC with sides 5 cm, 12 cm and 13 cm isrevolved about the side 12 cm. Find the volume of thesolid so obtained.[Hint :

Radius, r = 5 cm ; height, h = 12 cm & slant height, =13 cm ]75. If the triangle ABC in the question 19 above is revolved

about the side 5 cm, than find the volume of the solid soobtained. Find also the ratio of the volumes of the twosolids obtained in Question 19 and 20.[Hint :

Radius, r = 12 cm ; height, h = 5 cm & slant height ,= 13 cm]

76. The base radii of the two right circular cones of thesame height are in the ratio 3 : 5. Find the ratio of theirvolumes.[Hint : Let r1 and r2 be the radii of two cones ; v1 and v2be their respective volumes and h be their height.

Then, ]77. If h, c and v be the height, curved surface and volume of

a cone, show that[Hint : h = height of cone ; c = curved surface of cone =

; v = volume of cone =Substitute the values of LHS to obtain RHS]


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78. The radius of a spherical balloon increases from 7 cm to14 cm as air is being pumped into it. Find the ratio ofsurface areas of the balloon in the two cases.

[Hint : r1 = 7 cm & r2 = 14 cm and let S1 and S2 be thesurface areas of respective spheres.

]79. The diameter of the moon is approximately one fourth

of the diameter of the earth. Find the ratio of theirsurface areas.[Hint : Let d1 and d2 be the diameters of the moon andthe earth respectively and S1 and S2 be their respective

surface areas.80. A right circular cylinder just encloses a sphere of radius

r. Find(i) Surface area of the sphere,(ii) Curved surface area of the cylinder,(iii) Ratio of the areas obtained in (i) and (ii).

[Hint : Radius of cylinder = radius of sphere = rHeight of cylinder = 2 × radius of sphere = 2r]

81. A shot-put is a metallic sphere of radius 4.9 cm. If thedensity of the metal is 7.8 g per cm3, find the mass of

the shot-put. .[Hint : Mass of 1 cm3 of metal = 7.8 gMass of the shot put = volume of shotput × 7.8 g]

82. The diameter of the moon is approximately one-fourththe diameter of the earth. What fraction of the volumeof the earth is the volume of the moon ?[Hint : Let d1 and d2 be the diameters of the moon and

the earth respectively. Then, d1 = d2

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83. If the number of square centimetres of the surface of asphere is equal to the number of cubic centimeters in itsvolume what is the diameter of the sphere ?

[Hint :84. A cone and hemisphere have equal bases and equal

volumes. Find the ratio of their heights.85. Twenty seven solid iron spheres, each of radius r and

surface area S are melted to form a sphere with surfacearea S’. Find the (i) radius r’ of the new sphere, (ii) ratioof S and S’.[Hint : Volume of 27 solid iron sphere each of radius r= volume of new sphere of radius R.Þ R = 3r

S =

S’ =86. Length of a class-room is two times its height and

breadth is times its height. The cost ofwhite-washing the walls at the rate of Rs. 1.60 per m2 isRs. 179.20. Find the cost of tilling the floor at the rate ofRs. 6.75 per m2.

87. The dimensions of a rectangular box are in the ratio 2 : 3: 4 and the difference between the cost of covering itwith sheet of paper at the rate of Rs. 4 and Rs. 4.50 persquare metre is Rs. 416. Find the dimensions of the box.

88. Find the number of bricks, each measuring 25 cm × 12.5cm × 7.5 cm required to construct a wall 6 m long, 5 mhigh and 0.5 m thick, while the cement and sand mixtureoccupies 1/20 of the volume of the wall.

89. A class room is 7 m long, 6.5 m wide and 4 m high. It hasone door 3 m × 1.4 m and three windows, eachmeasuring 2 m × 1 m. The interior walls are to be colourwashed. The contractor charges Rs. 525 per sq. m. Findthe cost of colour washing.

90. A room is half as long again as it is broad. The cost ofcarpeting the room at Rs. 3.25 per m2 is Rs. 175.50 andthe cost of papering the walls at Rs. 1.40 per m2 is Rs.240.80. If 1 door and 2 windows occupy 8 m2, find thedimensions of the room.


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91. A wooden bookshelf has external dimensions as follows: Height = 110 cm, Depth = 25 cm, Breadth = 85 cm. Thethickness of the plank is 5 cm everywhere. The externalfaces are to be polished and the inner faces are to bepainted. If the rate of polishing is 20 paise per cm2 andthe rate of painting is 10 paise per cm2. Find the totalexpenses required for polishing and painting the surfaceof the bookshelf.

92. In fig. the shape of a solid csopper piece (made of twopieces with dimensions as shown in the figure) is shown.The face ABCDEFA is the uniform cross section. Assumethat the angle at A, B, C, D, E and F are right angles.Calculate the volume of the piece.

93. A plot of land in the form of a rectangle has a dimension240 m × 180 m. A drain let 10 m wide is dug all around it(on the outside) and the earth dug out is evenly spreadover the plot, increasing its surface level by 25 cm.Find the depth of the drain let.

94. A metallic sheet is of the rectangular shape withdimensions 48 cm × 36 cm. From each one of its corners,a square of 8 cm is cutoff. An open box is made of theremaining sheet. Find the volume of the box.

95. Water in a canal, 30 dm wide and 12 dm depp, is flowingwith a velocity of 20 km per hour. How much area will itirrigate in 30 min, if 9 cm of standing water is desired ?

96. A cylindrical road roller made of iron is 1 m wide. Itsinner diameter is 54 cm and thickness of the iron sheetrolled into the road roller is 9 cm. Find the weight of theroller it 1 c.c. of iron weights 8 g.

97. A solid cylinder has total surface area of 462 square cm.Its curved surface area is one-third of its total surfacearea. Find the volume of the cylinder (Take = 22/7)

98. A well with 10 m inside diameter is dug 14 m deep.Earth taken out of it is spread all around to a width of 5m to form an embankment. Find the height ofembankment.

99. A tent is of the shape of a right circular cylinder upto aheight of 3 metres and then becomes a right circularcone with a maximum height of 13.5 metres above theground. Calculate the cost of painting the inner side ofthe tent at rate of Rs. 2 per square metre, if the radius ofthe base is 14 metres.

100. A solid cube of side 7 cm is melted to make a coneof height 5 cm, find the radius of the base of the cone.

101. From a right circular cylinder with height 10 cm andradius of base 6 cm, a right circular cone of the sameheight and base is removed. Find the volume of theremaining solid.

102. The internal and external diameters of a hollowhemispherical vessel are 24 cm and 25 cm respectively.The cost to paint 1 cm2 surface is Rs. 0.05. Find the total

cost to paint the vessel all over.


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103. A wooden toy is in the form of a cone surmountedon a hemisphere. The diameter of the base of the coneis 6 cm and its height is 4 cm. Find the cost of paintingthe toy at the rate of Rs. 5 per 1000 cm2.

104. The front compound wall of a house is decorated bywooden spheres of diameter 21 cm, placed on smallsupports as shown in fig. Eight such spheres are used forthis purpose, and are to be painted silver. Each supportis a cylinder of radius 1.5 cm and height 7 cm and is tobe painted black. Find the cost of paint required if silverpaint costs 25 paise per cm2 and black paint costs 5paise per cm2.

105. A cylindrical container of radius 6 cm and height 15cm is filled with ice-cream. The whole ice-cream has tobe distributed to 10 children in equal cones withhemispherical tops. If the height of the conical portion isfour times the radius of its base, find the radius of theice-cream cone.

106. The dimensions of cuboid are in the ratio of 1 : 2 : 3and its total surface are is 88 m2. Find the dimensions.

107. Three cubes each of side 5 m are joined end to end.Find the surface are of the resulting cuboid.

108. A swimming pool is 20 m in length, 15 m in breadth,and 4 m in depth. Find the cost of cementing its floorand walls at the rate of Rs. 12 per square metre.

109. A cuboid has total surface area of 40 m2 and itslateral surface area is 26 m2. Find the area of its base.

110. The length of a cold storage is double its breadth. Itsheight is 3 metres. The area of its four walls (includingdoors) in 108 m2. Find its volume.

111. The sum of length, breadth and depth of a cuboid is19 cm and the length of its diagonal is 11 cm. Find thesurface area of the cuboid.

112. An open box is made of wood 3 cm thick. Itsexternal length, breadth and height are 1.48 m, 1.16 mand 8.3 dm. Find the cost of painting the inner surface atRs. 50 per sq. metre.

113. A cube of 9 cm edge is immersed completely in arectangular vessel containing water. If the dimensions ofthe base are 15 cm and 12 cm. Find the rise in waterlevel in the vessel.

114. A solid cube is cut into two cuboids of equalvolumes. Find the ratio of the total surface area of thegiven cube and that of one of the cuboids.

115. Three metal cubes whose edges measure 3 cm, 4 cmand 5 cm respectively are melted to form a single cube.Find its edge. Also, find the surface area of the newcube.

116. The area of the base of right circular cylinder is 154cm2 and its height is 15 cm. Find the volume of thecylinder.

117. The thickness of a hollow wooden cylinder is 2 cm. Itis 35 cm long and its inner radius is 12 cm. Find thevolume of the wood required to make the cylinder,assuming it is open at either end.

118. The radius and height of a cylinder are in the ratio 5: 7 and its volume is 550 cm3. Find its

radius.119. The volume of metallic cylindrical pipe is 748 cm3.Its length is 14 cm and its external radius is 9 cm. Find itsthickness.

120. The diameter of a cone is 14 cm and its slant heightis 9 cm. Find the area of its curved surface.

121. Find the total surface area of a cone, if its slantheight is 9 m and the radius of its base is 12 m.

122. The radius of a cone is 3 cm and vertical height is 4cm. Find the area of the curved surface.

123. The radius and slant height of a cone are in the ratio4 : 7. It its curved surface are is 792 m2, find its radius.

124. The lateral surface of a cylinder is equal to thecurved surface of a cone. If the radius be the same, findthe ratio of the height of the cylinder and slant height ofthe cone.

125. Find the volume of a right circular cone 1.02 m high,if the radius of its base is 28 cm.

126. The diameter of a right circular cone is 8 cm and its

volume is 48 . What it its height ?


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127. A right circular cone is 3.6 cm high and radius of itsbase is 1.6 cm. It is melted and recast into a right circularcone with radius of its base as 1.2 cm. Find its height.

128. A conical vessel whose internal radius is 5 cm andheight 24 cm is full of water. The water is emptied into acylindrical vessel with internal radius 10 cm. Find theheight to which the water rises.

129. A cone a cylinder are having the same base. Find theratio of their heights if their volumes are equal.

130. Find the surface are and total surface area of ahemisphere of radius 21 cm.

131. A sphere, a cylinder and a cone are of the sameradius and same height. Find the ratio of their curvedsurface.

132. Sow that the surface area of a sphere is the same asthat of the lateral surface of a right circular cylinder thatjust encloses the sphere.

133. The internal and external diameters of a hollowhemi-spherical vessel are 24 cm and 25 cm respectively.The cost of paint one sq. cm of the surface is 7 paise.Find the total cost to paint the vessel all over. (ignorethe area of edge).

134. Find the volume of a sphere whose surface area is154 square cm.

135. A solid sphere of radius 3 cm is melted and then castinto small spherical balls each of diameter 0.6 cm. Findthe number of balls thus obtained.

136. How many spherical bullets can be made out of asolid cube of lead whose edge measures 44 cm, eachbullet being 4 cm in diameter.

137. A solid lead ball of radius 7 cm was melted and thendrawn into a wire of diameter 0.2 cm. Find the length ofthe wire.

138. Write the lateral surface area of a cuboid havinglength units, breadth b units and height h. units.

139. Write the total surface area of a cuboid having threeedges of length as 10 cm, 5 cm and 3 cm.

140. Write the curved surface area of a right circularcylinder whose radius is 3 cm and height is 5 cm.

141. The volume of right cylinder having base radius 10cm is 600 cm3 . Find the height of the cylinder.

142. Write the curved surface area of a right circular conehaving radius 7 cm and slant height 10

cm.

143. Write the total surface area of a right circular solidcone having radius 10 cm and slant height 25

cm.144. Find the vertical height of a right circular cone

whose radius is 6 cm and slant height is 10 cm.145. Find the volume of a right circular cylinder having

radius 8 cm and height 10.5 cm.146. Write the volume of a right circular cone having

radius r and height h.147. Find the quantity of water in litres in a

hemispherical bowl of radius 21 cm. The bowl is

completely filled with water.148. The volume of a cuboid is 440 cm3 and the area of

its base is 88 m2. Find its height.149. The volume of cube is 1000 cm. Find its total surface

area.150. How many 3 metre cubes can be cut from a cuboid

measuring 18 m × 12 m × 9 m ?151. In fig, DP = BQ, ÐDPB = ÐBQD and ÐADP = ÐCBQ,

Show that DADP DCBQ.

[Hint : ÐDPB = ÐBQDÞ 1800 - ÐDPB = 1800 - ÐBQD ÞÐAPD = ÐCQB]


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152. and m are two parallel lines intersected byanother pair of parallel lines p and q. Show that DABC

DCDA.

[Hint: (i) ÐBAC = ÐDCA(ii) ÐACB = ÐCAD(iii) AC = CA

153. Ram wishes to determine the distance between twoobjects A and B, but there is an obstacle between thesetwo objects as shown in fig, which prevents his frommaking a direct measurement. He devises an ingeniousway to overcome this difficultly. First, he fixes a pole at aconvenient point O so that from O, both A and B arevisible. Then, he fixes another pole at the point D on theline AO (produced) such that AO = DO. In a similar way,he fixes a third pole at the point C on the line BO(produced) such that BO = CO. Then he measures CDand finds that CD = 170 m. Prove that the distancebetween the object A and B is also 170 m.

154. In right triangle ABC, right angled at C, M is themid-point of hypotenuse AB. C is joined to M andproduced to a point D such that DM = CM. Point D isjoined to point B. Show that :(i) DAMC DBMD(ii) ÐDBC is a right angle.(iii) DDBC DACB]

(iv) CM = AB.

155. In fig AB||QR, BC||PR and AC = PQ. Prove thatDABC DQRP.

[Hint : ÐBAC = ÐRQP (alternate interior angles)AC = PQÐBCA = ÐRPQ (alternate exterior angles)]

156. E and F are respectively the mid-points of equalsides AB and AC of DABC. Show that BF = CE.

[Hint : To show BF = CE, prove DABF DACE.]157. In fig, BL AC, MC LN, AL = CN and BL = CM.

Prove that DABC DNML.

[Hint : By SAS congruence rules, DALB DNCMÞ AB = NM and ÐLAB = ÐCNMIn DABC and DNML, AB = NM, ÐCAB = ÐLNM,AL = CN Þ AL + LC =CN + LC Þ AC = NL.]

158. In an isosceles triangle ABC with AB = AC, D and Eare points on BC such that BE = CD. Show that AD = AE.


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159. If two isosceles triangle have a common base, theline joining their vertices bisects them at right angles.[Hint : There can be two possible situations :By SSS congruence rule, DABD DACD ÞÐ1 = Ð2

By SSS congruence rule, DBAE DCAE ÞÐ3 = Ð4 &Be = EC.]

160. In an isosceles triangle ABC, with AB = AC, thebisectors of ÐB and ÐC intersect each other at O. Join Ato O. Show that :(i) OB = OC (ii) AO bisects ÐA

[Hint : (i) AB = AC ÞÐB = ÐC Þ ÐB = ÐC ÞÐOBC = ÐOCBÞ OB = OC(ii) By SAS congruence rule, DAOB DAOCÞÐOAB = ÐOAC]

161. Prove that any two sides of a triangle are togethergreater than twice the median drawn to the third side.[Hint :

In DABC, AD is the median. Produced AD to E such thatAD = DE. Join EC.DADB DEDC Þ AB = ECIn DAEC, AC + EC > AE Þ AC + AB > 2AD]

162. DABC and DDBC are two isosceles triangles on thesame base BC and vertices A and D are on the same sideof BC. If AD is extended to intersect BC at P, show that :

(i) DABD DACD(ii) DABP DACP(iii) AP bisects ÐA as well as ÐD.(iv) AP is the perpendicular bisector of BC.[Hint : (i) By SSS congruence rule, DABD DACD(ii) ÞÐBAP = ÐCAPIn DABP & DACP, AB = AC, AP = AP, ÐBAP = ÐCAP(iii) To show AP bisects ÐD, prove DDBP DCDP]

163. A point O is taken inside an equilateral four sidesfigure ABCD such that its distances from the angularpoints D and B are equal. Show that AO and OC are inone such the same straight line.[Hint :

DAOD DAOD ÞÐ1 = Ð2DDOC DBOC ÞÐ3 = Ð4Ð1 + Ð2 + Ð3 + Ð4 = 3600

164. Two sides AB and BC and median AM of one triangleABC are respectively equal to sides PQ and QR andmedian PN of DPQR. Show that :(i) DABM DPQN(ii) DABC DPQR

[Hint : (i) By SSS congruence rule, DABM DPQN(ii) By SAS congruence rule, DABC DPQR]


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165. In a right angled triangle, one acute angle is doublethe other. Prove that the hypotenuse is double thesmallest side.

166. ABC is a triangle in which altitudes BE and CF tosides AC and AB are equal. Show that :(i) DABE DACF (ii) AB = AC, i.e., ABC is an isoscelestriangle.

167. In fig. DABC is right angled at C, DPQR is right angledat R. If AB = PQ and BC = PR, prove that DACP DQRB.

168. In fig. AB = AC, AD BC, BE = DE and CF = DF.Prove that :(i) DABE DACF(ii) ÐBAE = ÐCAF

[Hint : By RHS congruence rule, DADB DADCÞ BD = DC and ÐB = ÐC

Now, BD = DC Þ BD = BC Þ BE = FC]

169. In fig, ÐAOB = ÐPOQ = 900, OB = OQ and AB = PQ.Prove that :(i) DOAB DOPQ.(ii) Ð1 = Ð2

170. Show that in a right angled triangle, the hypotenuseis the longest side.

[Hint : In a right angled DABC with ÐC = 900, ÐA + ÐB =900ÞÐA < 900, ÐB < 900Þ AB > BC, AB > AC]

171. Show that the difference of any two sides of atriangle is less than the third side.

172. In fig. ÐB < ÐA and ÐC < ÐD. Show that AD < BC.[Hint: ÐB < ÐA Þ OA < OB ...(i)ÐC < ÐD Þ OD < OC ...(ii)Add (i) & (ii)]

173. In fig. AP and PR > PQ. Show that AR > AQ.

174. In fig. PR > PQ and PS bisects ÐQPR. Prove thatÐPSR > ÐPSQ.

[Hint : PR > PQ ÞÐQ > ÐRand ÐQPS = ÐRPSÞÐQ + ÐQPS > ÐR + ÐRPS ÞÐPSR > ÐPSQ]


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175. In fig. AB > AC, PB and PC are bisectors of ÐB andÐC respectively. Show that PB > PC.

176. In fig. O is an interior point of DABC. BO meets AC atD. Show that OB + OC < AB + AC.

177. In fig, ABCD is a quadrilateral in which diagonals ACand BD intersect at O. Show that 2(AC + BD) > AB + BC +CD + DA.

[Hint : OA + OB < ABOB + OC > BCOC + OD > CDOD + OA > DA]

178. Whish of the following pairs of triangles arecongruent ?(a) DABC and DDEF in which : BC = EF, AC = DF and ÐC =ÐF.(b) DABC and DPQR in which : AB = PQ BC = QR and ÐC =ÐR.(c) DABC and DLMN in which : ÐA = ÐL = 900, AB = LM,ÐC = 400 and ÐM = 500.(d) DABC and DDEF in which : ÐB = ÐE = 900 and AC =DF.

179. Answer the following as per the exact requirement :(a) In Ds ABC and PQR, AB = PQ, AC = PR and ÐBAC =ÐQPR.Here, DABC DPQR.Justify the statement by writing the congruencecriteria applicable in this situation.(b) In fig. ÐBAC = ÐQRP.Justify that DABC DRQP

180. In DABC, AB = AC. OB and OC are bisectors of ÐBand ÐC respectively. Show that OB = OC.

181. In fig. Ð1 > Ð2. Show that AB > AC.

182. In DABC, we have, ÐA > ÐB > ÐC, then determinethe shortest and the longest side of the triangle.

183. If DABC DPQR, ÐB = 400 and ÐC = 950, find ÐP.184. In DABC, AB = BC = 5 cm and ÐA = 550, find DB.185. State the angle angle-side congruence criteria for

triangles.186. In fig, AB = AC and ÐACD = 1150. Find ÐA.

187. In DABC, BC = AC and ÐB = 640, find ÐC.188. In DPQR, ÐP = 500 and ÐR = 700, Name (i) the

shortest side (ii) the longest side of the triangle.


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189. In the given fig. ABCD is a square and DPAB is anequilateral triangle.

(i) Prove that DAPD DBPC.(ii) Show that ÐDPC = 150.

190. In the given fig. in DABC, ÐB = 900. if ABPQ and ACRSare squares,prove that :(i) DACQ DABS.(ii) CQ = BS.

191. Squares ABPQ and ADRS are drawn on the sides ABand AD of a parallelogram ABCD. Prove that:(i) ÐSAQ = ÐABC(ii) SQ = AC.

192. In the given fig, ABCD is a square and P, Q, R arepoints on AB, BC and CD respectively such that AP = BQ= CR and ÐPQR = 900. Prove that: (i) PB = QC, (ii) PQ =QR (iii) ÐQPR = 450.

193. In the given fig, ABCD is a square, EF||BD and R isthe mid-point of EF. Prove that :(i) BE = DF(ii) AR bisects ÐBAD(iii) If AR is produced, it will pass through C

194. In a DABC, AB = AC and BC is produced to D. From D,DE is drawn perpendicular to BA produced and DF is -drawn perpendicular to AC produced. Prove that BDbisects ÐEDF.

195. Prove that the perimeter of a triangle is greater thanthe sum of its three medians.

196. In the adjoining figure, prove that :(i) AB + BC + CD > DA(ii) AB + BC + CD + DA > 2AC(iii) AB + BC + CD + DA > 2BD(iv) AB + BC + CD + DA > AC + BD

197. In the adjoining figure, O is the centre of a circle, XYis a diameter and XZ is a chord. Prove that XY > XZ.


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198. In the given figure, AD = AB and AE bisects ÐA.Prove that :(i) BE = ED(ii) ÐABD > ÐBCA.

199. In the given fig, the line segments AB and CDintersect at a point M in such a way that AM = MD andCM = MB. Prove that, AC = BD but AC many not beparallel to BD.

200. In the given fig. AY ZY and BY XY such thatAY = ZY and BY = XY. Prove that AB = ZX.

201. If the bisector of the exterior vertical angle of atriangle is parallel to the base, show that the triangle isisosceles.

1. In each of the following figures, find the value of x :

203. In each of the following figures, find the value of x:

204. In the given fig, BD || CE; AC = BC, DABD = 200 andÐECF = 700. Find ÐGAC.

205. In the given figure, AB || CD and CA = CE. Find thevalues of x, y and z.

206. In the given figure, AB = AD; CD; ÐA = 420 and ÐC =1080, find ÐABC.


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207. In the given figure, side BA of DABC has beenproduced to D such that CD = CA and side CB has beenproduced to E. If ÐBAC = 1060 and ÐABE = 1280 findÐBCD.

208. In the given figure, AB = BC and AC = CD. Show thatÐBAD : ÐADB = 3 : 1.

209. In the given figure, AD is the internal bisector of ÐAand CE || DA. If CE meets BA produces at E, prove thatDCAE is isosceles.

210. In the given figure, AD bisects ÐA. Arrange AB, BDand DC in ascending order.

211. In the given fig. AB = AC. Prove that : AF > AE.

212. In the given figure, side AB of DABC is produced to Dsuch that BD = BC.IF ÐA = 600and ÐB = 500 prove that :(i) AD > CD(ii) AD > AC

213. In the given figure, AD bisect ÐA. If ÐB = 600, Ð =400, then arrange AB, BD and DC in ascending order oftheir lengths.

214. Find the area of a triangle whose sides arerespectively 150 cm, 120 cm and 200 cm

215. Find the area of a triangle whose sides are 9 cm, 12cm and 15 cm

216. Find the area of a triangle, two sides of which are 18cm and 10 cm and the perimeter is 42 cm.

217. In a DABC, AB = 15cm, BC = 13 cm and AC = 14 cm.Find the area of DABC and hence its altitude on AC.

218. The perimeters of a right triangle is 450 m. If itssides are in the ratio 13 : 12 : 5. Find the area of thetriangle.219. Find the area of a quadrilateral ABCD in which AB= 3 cm, BC = 4 cm, CD = 4 cm, DA = 5 cm and AC = 5 cm.

220. The sides of a quadrangular field, taken is order are26m, 27m, 7m, and 24m respectively. The anglecontained by the last two sides is a right angle. Find itsArea.

221. Two parallel sides of a trapezium are 60 cm and 77cm and other sides are 25 cm and 26 cm. Find the areaof the trapezium.


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222. Find the area of DABC and DACD in given figure.

223. Find the area of rhombus, if perimeter is 120 m andlonger diagonal is 48 m.

224. Using Hero's formula find the area of an isoscelestriangle whose one of the equal sides is 16 cm and thirdside is 10 cm.

225. The perimeter of a right triangle is 144 cm and itshypotenuse measures 65 cm. Find the lengths of othersides and calculate its area. Verify the result using Hero'sformula.

226. The base of an isosceles triangle is 14 cm and one ofits equal sides is 12 cm. Find its area using Hero'sformula.

227. The sides of a triangle are of lengths 10 cm, 15 cmand 15 cm. Find the length of the altitude drawn on theside with length 15 cm.

228. An isosceles right triangle has an area 200 cm2.What is the length of its hypotenuse?

229. The perimeter of a right triangles is 12 cm and itshypotenuse is of length 5 cm. Fine the other two sidesand calculate its area.

230. The sides of a triangle are of lengths 8 cm, 15 cmand 17 cm. Find the length of the altitude drawn on theside with length 17 cm.

231. The base of an isosceles triangle measures 24 cmand its area is 192 cm2. Find its perimeter.

232. Find the area of an isosceles right-angled triangle,each of whose equal sides measures 10 cm. (Take =1.414)

233. Find the base of an isosceles triangle whose area is12 cm2 and one of the equal sides in 5 cm.

234. Find the percentage increase in the area of atriangle if its each side is doubled.

235. The lengths of the sides of triangle ABC are in theratio 4 : 3 : 5, and its perimeter is 144 cm. Find theheight corresponding to the longest side.

236. The dimensions of a cuboid are in the ratioof 1 : 2 : 3 and its total surface area is 88 m2. Find thedimension.

237. A swimming pool is 20 m in length, 15 m in breadth,and 4 m in depth. Find the cost of cementing its floorand walls at the rate of Rs 12 per square meter.

238. The floor of a rectangular hall has a perimeter 250m. If the cost of painting the four walls at the rate of 10per m2 is Rs 15000. Find the height of the hall.

239. The sum of length, breadth and depth of a cuboid is19 cm and the length of its diagonal is 11 cm. Find thesurface area of the cuboid.

240. Three cubes whose edges measure 3 cm, 4 cm and 5cm respectively to form a single cube. Find its edge.Also, find the surface area of the new cube.

241. A reservoir is in the form of a rectangularparallelopiped (cuboid). Its length is 20 m. If 18 kl ofwater is removed from the reservoir, the water levelgoes down by 15 cm. Find the width of the reservoir (1kl = 1 m3).

242. The outer dimensions of a closed wooden box are10 cm by 8 cm by 7 cm. Thickness of the wood is 1 cm.Find the total cost of wood required to make box if 1cm3 of wood cost Rs 2.00.

243. Water flows in a tank 150 m × 100 m at the base,through a pipe whose crosssection is 2 dm by 1.5 dm atthe speed of 15 km per hour. In what time, will thewater be 3 metres deep.

244. An iron pipe 20 cm long has exterior diameter equalto 25 cm. If the thickness of the pipe is 1 cm, find thewhole surface of the pipe.

245. The diameter of a roller 120 cm long is 84 cm. If ittakes 500 complete revolutions to level a playground,determine the cost of levelling it at the rate of 30 paiseper square metre.

246. The thickness of a hollow wooden cylinder is 2 cm. Itis 35 cm long and its inner radius is 12 cm. Find thevolume of the wood required to make the cylinder,assuming it is open at either end.

247. The circumference of the base of a cylindrical vesselis 132 cm and its height is 25 cm. How many liters ofwater can it hold ?

248. The volume of a cylinder is 448 p cm3 and height 7cm. Find its lateral surface area and total surface area.

249. The volume of metallic cylindrical pipe is 748 cm3.Its length is 14 cm and its external radius is 9 cm. Find itsthickness.


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250. The circumference of the base of a 10 m high conicaltent is 44 metres. Calculate the length of canvas used inmaking the tent if width of canvas is 2 m. (Use p = 22/7).

251. The base radii of two right circular cones of thesame height are in the ratio 3 : 5. Find the ratio of theirvolumes.

252. A right circular cone is 3.6 cm high and radius of itsbase is 1.6 cm. It is melted and recast into a right circularcone with radius of its base as 1.2 cm. Find its height.

253. A solid cube of side 7 cm is melted to make a coneof height 5 cm, find the radius of the base of the cone.

254. The radius and height of a cone are in the ratio 3 : 4.If its volume is 301.44 cm3, what is its radius ? What is itsslant height ? (Take p = 3.14)

255. The internal and external diameters of a hollowhemi-spherical vessel are 24 cm and 25 cm respectively.The cost of paint one sq. cm of the surface is 7 paise.Find the total cost to paint the vessel all over. (ignorethe area of edge).

256. A toy is in the shape of a right circular cyling with ahemisphere on one end and a cone on the other. Theheight and radius of the cylindrical part are 13 cm and 5cm respectively. The radii of the hemispherical andconical parts are the same as that of the cylindrical part.Calculate the surface area of the toy if height of theconical part is 12 cm.

257. Find the volume of a sphere whose surface area is154 square cm.

258. A solid sphere of radius 3 cm is melted and then castinto small spherical balls each of diameter 0.6 cm. Findthe number of balls thus obtained.

259. How many spherical bullets can be made out of asolid cube of lead whose edge measures 44 cm, eachbullet being 4 cm in diameter.

260. Three solid spheres of iron whose diameters are 2cm, 12 cm and 16 cm, respectively, are melted into asingle solid sphere. Find the radius of the solid sphere.

261. A sphere of diameter 6 cm is dropped in a rightcircular cylindrical vessel partly filled with water. Thediameter of the cylindrical vessel is 12 cm. If the sphereis completely submerged in water, by how much will thelevel of water rise in the cylindrical vessel ?

262. A spherical canon ball, 28 cm in diameter is meltedand cast into a right circular conical mould, the base ofwhich is 35 cm in diameter. Find the height of the cone,correct to one placed of decimal.

263. The base of right prism is an equilateral triangle ofside 5 cm. If the lateral surface area of the prism is 750cm2, find its volume.

264. A cuboidal oil tin is 30 cm by 40 cm by 50 cm. Findthe cost of the tin required for making 20 such tins if thecost of tin sheet is Rs 20 per square metre.

265. Length of a class-room is two times its height and its

breadth is times its height. The cost ofwhite-washing the walls at the rate of Rs 1.60 per m2 isRs 179.20. Find the cost of tiling the floor at the rate ofRs 6.75 per m2.

266. A room is half as long again as it is broad. The cost ofcarpeting the room at Rs 3.25 per m2 is Rs 175.50 andthe cost of papering the walls at Rs 1.40 per m2 is Rs240.80. If 1 door and 2 windows occupy 8 m2, find thedimensions of the room.

267. The cost of papering four walls of a room at 70 paiseper square metre is Rs 157.50. The height of the room is5 meters. Find the length and the breadth of the room ifthey are in the ratio 4 : 1.

268. A plot of land in the form of a rectangle has adimension 240 m × 180 m. A drainlet 10 m wide is dugall around it (on the outside) and the earth dug out isevenly spread over the plot, increasing its surface levelby 25 cm. Find the depth of the drainlet.

269. An agricultural field is in the form of a rectangle oflength 20 m and width 14 m. A pit 6 m long, 3 m wideand 2.5 m deep is dug in a corner of the field and theearth taken out of the pit is spread uniformly over theremaining area of the field. Find the extent to which thelevel of the field has been raised.

270. A rectangular tank is 225 m by 162 m at the base.With what speed must water flow into it through andaperture 60 cm by 45 cm that the level may be raised 20cm in 5 hours?

271. The external length, breadth and height of a closedrectangular wooden box are 18 cm, 10 cm and 6 cmrespectively and thickness of wood is 1/2 cm. When thebox is empty, it weight 15 kg and when filled with sand itweighs 100 kg. Find the weight of the cubic cm of woodand cubic cm of sand.

272. A rectangular sheet of paper 44 cm × 18 cm is rolledalong its length and a cylinder is formed. Find the radiusof the cylinder.


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273. A metal pipe is 77 cm long. The inner diameter of across section is 4 cm, the outer diameter being 4.2 cm.Find its(i) inner curved surface area(ii) outer curved surface area.(iii) total surface area

274. A solid cylinder has total surface area of 462 squarecm. Its curved surface area is one-third of its totalsurface area. Find the volume of the cylinder. (Take p =22/7)

275. The difference between outside and inside surfaceof a cylindrical metallic pipe 14 cm long is 44 cm2. If thepipe is made of 99 cu centimeters of metal, find theouter and inner radii of the pipe.

276. A lead pencil consists of a cylinder of wood with asolid cylinder of graphite filled into it. The diameter ofthe pencil is 7 mm, the diameter of the graphite is 1 mmand the length of the pencil is 10 cm. Calculate theweight of the whole pencil, if the specific gravity of thewood is 0.7 gm/cm3 and that of the graphite is 2.1gm/cm3.

277. The radius and height of a cone are in the ratio 4 : 3.The area of the base is 154 cm2. Find the area of thecurved surface.

278. A tent is of the shape of a right circular cylinder uptoa height of 3 metres and then becomes a right circularcone with a maximum height of 13.5 metres above theground. Calculate the cost of painting the inner side ofthe tent at the rate of Rs 2 per square metre, if theradius of the base is 14 metres.

279. If h, C, V are respectively the height , the curvedsurface and the volume of a cone, prove that 3pVh3 -C2h2 + 9V2 = 0

280. A cone of height 24 cm has a curved surface area550 cm2. Find its volume. (Take p = 22/7).

281. A conical tent is 9 m high and the radius of its base is12 m.(i) What is the cost of the canvas required to make it, if asquare metre canvas costs Rs 10 ?(ii) How many persons can be accommodated in thetent, if each person requires 2 square metre on theground and 15 m3 of space to breathe in ?

282. A wooden toy is in the form of a cone surmountedon a hemisphere. The diameter of the base of the coneis 6 cm and its height is 4 cm. Find the cost of paintingthe toy at the rate of Rs 5 pr 1000 cm2.

283. The diameter of a sphere is decreased by 25%. Bywhat percent its curved surface areadecrease ?

284. A cylindrical container of radius 6 cm and height 15cm is filled with ice-cream. The whole ice-cream has tobe distributed to 10 children in equal cones withhemispherical tops. If the height of the conical portion isfour times the radius of its base, find the radius of theice-cream cone.

285. A solid wooden toy is in the shape of a right circularcone mounted on a hemisphere. If the radius of thehemisphere is 4.2 cm and the total height of the toy is10.2 cm, find the volume of the wooden toy.

286. A vessel is in the form of a hemispherical bowlmounted by a hollow cylinder. The diameter of thesphere is 14 cm and the total height of the vessel is 13cm. Find its capacity. (Take p = 22/7)

287. A solid is in the form of a cylinder withhemispherical ends. The total height of the solid is 19cm and the diameter of the cylinder is 7 cm. Find thevolume and total surface area of the solid. (use p =22/7).288. Find the surface area of a chalk box whose length,breadth and height are 16 cm, 8 cm and 6 cm,respectively.

289. Three cubes each of side 5 cm are joined end to end.Find the surface area of the resulting cuboid.S

290. Find the area of the four walls of a room whoselength is 6m, breadth 5m and height 4m. Also find thecost of white-washing the walls, if the rate of whitewashing is Rs. 5 per square meter (Door, Windows andother openings ignored).

291. The length of a cold storage is double its breadth. Itsheight is 3 meters. The area of its four walls (includingdoors) is 108 m2. Find its volume.

292. The volume of a cuboid is 440 cm3 and the area ofits base is 88 cm2. Find its height.

293. The volume of a cube is 1,000 cm. Find its totalsurface area.

294. The curved surface area of a right circular cylinder ofheight 14 cm is 88 cm2. Find the diameter of the base ofthe cylinder.

295. The ratio between the curved surface area and thetotal surface area of a right circular cylinder is 1 : 2. Findthe ratio between the height and radius of the cylinder.

296. Find the volume of a right circular cylinder, if theradius (r) of its base and height (h) are 7 cm and 15 cmrespectively.


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297. The area of the base of a right circular cylinder is154 cm2 and its height is 12 cm. Find the volume of thecylinder.

298. The area of the base of a right circular cone is 314cm2 and its height is 15 cm. Find the volume of the cone.

299. Find the surface area of a sphere of radius 7 cm.300. Find the surface area and total surface area of a

hemisphere of radius 21 cm.301. Find the volume of a sphere of radius 7 cm.302. Find the volume of hemisphere of radius 3.5 cm.303. Find the area of the base of a right triangular prism

having volume 672 cm2 and height 8 cm.304. If the ratio of three angles of a triangle is1 : 2 : 3,

find the angles.305. In the fig. l || m and n is transversal. PO and QO are

angle bisectors. Prove that ÐPOQ = 900

306. If the angles of a triangle are in the ratio 2 : 3 : 4,determine the three angles

307. The sum of two angles of a triangle is 950 and theirdifference is 250. Find all the three angles of the triangle

308. The side BC of a triangle ABC is produced to D. Thebisector of the ÐA meets BC in L. Prove that ÐABC +ÐACD = 2 ÐALC

309. The sides BC, CA and AB of DABC, are produced Inorder, forming exterior angles ÐACD, ÐBAE andÐCBF. Show that ÐACD + ÐBAE + ÐCBF = 360

310. Sides BC, CA and BA of the DABC are produced to D,E, F, respectively. If ÐACD = 1100 and ÐEAF = 1300. Findall the three angles of the triangle.

311. In the adjoining figure, find the value of , ÐA + ÐB +ÐC + ÐD + ÐE + ÐF.

312. In a DABC, the angle bisectors of the ÐABC and theÐACB meet at O. If ÐABC = 800, find ÐBOC

313. If the sides AB and AC of DABC are produced (Fig.)and ÐDBC < ÐBCE.

314. Prove that hypotenuse is the longest side in aright-angled-triangle.

315. Prove that the sum of the three sides of a triangleis greater than the sum of its three medians.

316. In Fig. the sides BA and CA have been producedsuch that BA = AD and CA = AE. Prove that segment DE|| side BC.

317. In Fig. PA ^ AB, QB ^ AB and PA = QB. If PQintersects AB at O , show that O is the mid-point of AB aswell as that of PQ.


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318. In the given figure, find the value of xº and yº

319. Prove that any two sides of a triangle are togethergreater than twice the median drawn to the third side

320. In the Fig. AP is the shortest line segment that canbe drawn from A to line m. It PR > PQ, prove that AR >AQ.

321. In quad. PQRS, (Fig.)

Prove that(i) PQ + QR + RS + SP > PR + QS(ii) PQ + QR + RS + SP < 2 (PR + QS)

322. In the Fig. PQ = PR and S is a point on PR, prove thatRS < QS.

323. In D ABC, AC > AB and D is the point on AC such thatAB = ADProve that CD < BC

324. In the given Fig. T is a point on side QR of DPQR andS is a point such that TR = TS. Prove that PQ + PR > QS.

325. Prove that each angle of an equilateral triangle is600

326. In Fig. If ÐAOC + ÐBOD = 700, find ÐCOD.

327. In Fig. ÐDFP, ÐEDQ and ÐFER are exterior angles ofD DEF. Prove thatÐ DFP + Ð EDQ + Ð FER = 3600

328. In fig. AB > AC and D is any point on side BC of DABC. Prove that AB > AD.


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329. In the given figure, PQ = PR. S is any point on theside PR. Prove that : RS < QS.

PR and QS are the diagonals of a quadrilateral. PQRS.Prove thatPQ + QR + RS + SP > PR + QS

330. In a parallelogram ABCD, two points P and Q aretaken on its diagonal BD such that DP = BQ. Prove thatPQ and AC bisect each other.

331. (1) Prove that two triangles are congruent if anytwo angles and the included side of one triangle arerespectively equal to any two angles and the includedside of the second triangles.(ii) Using the above therorem, prove thatCF = AD inthe given figure in which E is the mid-point of ACand CF drawn parallel to DB.

332. An exterior angle of a triangle is 120º. One of theinterior opposite angle is 35º. Find theother two angles

333. A triangle ABC is right angled at A. AL isperpendicular to BC. Prove that ÐBAL = ÐBCA.

334. In the Fig. PS is the bisector of the ÐP andPT ^ QR, then show that

ÐTPS = (ÐQ - ÐR)335. In the Fig. AM ^ BC and AN is th angle bisector of

ÐA if ÐB = 600 and ÐC = 500, find ÐMAN.

336. In the given figure, AM ^ BC and AN is the bisectorof ÐBAC. If ÐB = 700 and ÐC = 350, find ÐMAN.

337. In the figure find the value of x0.

338. S the point in the interior of DPQR. Prove that SQ +SR < PQ + PR.


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339. Prove that the sum of the three altitudes of atriangle is less than the sum of the three sides of thetriangle.

340. In the Fig. Q is a point on side SR of D PSR such thatPQ = PR, prove that PS > PR.

341. In D ABC, AD is the median and D lies on BC, provethat AB + AC > 2AD.

342. In DABC, AC > AB (Fig.) and BD and CD are anglebisectors of ÐB and ÐC respectively.Prove that DC > BD.

343. Prove that the medians of an equilateral triangle areequal.

344. Angles A, B, C of a triangle ABC are equal to eachother. Prove that DABC is equilateral.

345. ABCD is a square, X and Y are points on sides AD andBC respectively such that AY = BX. Prove thatBY = AX and ÐBAY = ÐABX.

346. In the given figure, DABC is an equilateral trianglethe length of whose side is equal to 10 cm and DDBC isright-angled at D and BD = 8 cm. Find the area of the

shaded region. Take = 1.732.

347. Calculate the area of the triangle whose sides are 18cm, 24 cm and 30 cm in length. Also, find the length ofthe altitude corresponding to the smallest side of thetriangle.

348. The sides of a triangle are 10 cm, 24 cm and 26cm. Find its area and the longest altitude.

349. Two sides of a triangular field are 85 m and 154 m inlength, and its perimeter is 324 cm. Find (i) the area ofthe field, and (ii) the length of the perpendicular fromthe opposite vertex on the side measuring 154 cm.

350. The sides of a triangular field are 165 cm, 143 cmand 154 cm. Find the cost of ploughing it at 12 paise persq. m.

351. The base of an isosceles triangle measures 80 cmand its area is 360 cm2. Find the perimeter of thetriangle

352. The perimeter of an isosceles triangle is 42 cm and

its base is times each of the equal sides. Find (i)the length of each side of the triangle, (ii) the area of thetriangle, and (iii) the height of the triangle.

353. The perimeter of a right angle triangle is 40 cm. Itshypotenuse is 17 cm. Find the sides containing the rightangle. Also find the area of the triangle.

354. Find the area and perimeter of an isoscelesright-angled triangle, each of whose equal sidesmeasures 10 cm.

Take = 1.414.355. The area of a square field is 8 hectares. How long

would a man take to cross its diagonal by walking at therate of 4 km per hour ?

356. A rhombus shaped field has green grass for 18 cowsto graze. If each side of the rhombus is 30 m and itslonger diagonal is 48 m, how much area of grass fieldwill each cow be getting ?

357. There is a slide in a park. One of its side walls hasbeen painted in some colour with a message “KEEP THEPARK GREEN AND CLEAN”. If the sides of the wall are 15m, 11 m and 6 m, find the area painted in colour.

358. Find the area of a triangle two sides of which are 18cmand 10cm and the perimeter is 42 cm.


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359. An isosceles triangle has perimeter 30cm and each ofthe equal sides is 12cm. Find the area of the triangle.

360. A park, in the shape of a quadrilateral ABCD, has DC =900, AB = 9m, BC = 12m, CD = 5m and AD = 8m. Howmuch area does it occupy ?

361. Find the area of a quadrilateral ABCD in which AB =3cm, BC = 4cm, CD = 4cm, DA = 5cm and AC = 5CM.

362. A rhombus shaped field has green grass for 18 cows tograze. If each side of the rhombus is 30m and itslonger diagonal in 48m, how much area of grass fieldwill each cow be getting ?

363. An umbrella is made by stitching 10 triangular piecesof cloth of two different colours, each piece measuring20cm, 50 cm and 50 cm. How much cloth of eachcolour is required for the umbrella ?

364. A kite in the shape of square with a diagonal 32 cmand an isosceles triangle of base 8cm and sides 6cmeach is to be made of three different shades as shownin fig. How much paper of each shade has been usedin it ?

365. A plastic box 1.5 m long, 1.25 m wide and 65 cm deepis to be made. It is opened at the top. Ignoring thethickness of the plastic sheet, determine :

(i) The area of the sheet required for making the box.(ii) The cost of sheet for it, if a sheet measuring 1 m2

costs Rs. 20.366. The length, breadth and height of a room are 5 m, 4 m

and 3 m respectively. Find the cost of white washingthe walls of the room and the ceiling at the rate of Rs.7.50 per m2.

367. A cubical box has each edge 10 cm and anothercuboidal box is 12.5 cm long, 10 cm wide and 8 cmhigh.

(i) Which ox has the greater lateral surface area and byhow much ?

(ii) Which box has the smaller total surface area and byhow much ?

368. A small indoor greenhouse (herbarium) is madeentirely of glass panes (including base) held togetherwith tape. It is 30 cm long, 25 cm wide and 25 cm high.

(i) What is the area of the glass ?(ii) How much of tape is needed for all the 12 edges ?369. It is required to make a closed cylindrical tank of

height 1 m and base diameter 140 cm from a metalsheet. How many square metres of the sheet arerequired for the same ?

370. A metal pipe is 77 cm long. The inner diameter of across section is 4 cm, the outer diameter being 4.4 cm.Find its :

(i) inner curved surface area,(ii) outer curved surface area,(iii) total surface area.371. A cylindrical pillar is 50 cm is diameter and 3.5 m in

height. Find the cost of painting the curved surface ofthe pillar at the rate of Rs. 12.50 per m2.

372. Curved surface area of a right circular cylinder is 4.4m2. If the radius of the base of the cylinder is 0.7 m,find its height.

373. The inner diameter of a circular well is 3.5 m. It is 10 mdeep. Find

(i) its inner curved surface area,(ii) the cost of plastering this curved surface at the rate of

Rs 40 per m2374. Find the total surface area of a cone, if its slant height

is 21 m and diameter of its base is 24 m.375. A joker’s cap is in the form of a right circular cone of

base radius 7 cm and height 24 cm. Find the area ofthe sheet required to make 10 such caps.

376. A bus stop is barricaded from the remaining part ofthe road, by using 50 hollow cones made of recycledcardboard, Each cone has a base diameter of 40 cmand height 1 m. If the outer side of each of the conesis to be painted and the cost of painting is Rs 12 perm2, what will be the cost of painting all these cones ?

(Use = 3.14 and take = 1.02)377. Find the surface area of a sphere of radius :(i) 10.5 cm (ii) 5.6 cm (iii) 14 cm


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378. A hemispherical bowl made of brass has innerdiameter 10.5 cm. Find the cost of tin-plating it on theinside at the rate of Rs. 16 per 100 cm2.

379. A cuboidal water tank is 6 m long, 5 m wide and 4.5 mdeep. How many litres of water can it hold

? (1 m3 = 1000 )380. Find the cost of digging a cuboidal pit 8 m long, 6 m

broad and 3 m deep at the rate of Rs. 30 per m3.381. A godown measures 40 m × 25 m × 10 m. Find the

maximum number of wooden crates each measuring1.5 m × 1.25 m × 0.5 m that can be stored in thegodown.

382. A river 3 m deep and 40 m wide is flowing at the rateof 2 km per hour. How much water will fall into thesea in a minute ?

383. A soft drink is available in two packs - (i) a tin can witha rectangular base of length 5 cm and width 4 cm,having a height of 15 cm and (ii) a plastic cylinder withcircular base of diameter 7 cm and height 10 cm.Which container has greater capacity and by howmuch ?

384. If the lateral surface of a cylinder is 94.2 cm2 and itsheight is 5 cm, then find

(i) radius of its base (ii) its volume. (Use = 3.14)385. A lead pencil consist of a cylinder of wood with a solid

cylinder of graphite filled in the interior. The diameterof the pencil is 7 mm and the diameter of the graphiteis 1 mm. If the length of the pencil is 14 cm, find thevolume of the wood and that of the graphite.

386. A patient in a hospital is given soup daily in acylindrical bowl of diameter 7 cm. If the bowl is filledwith soup to a height of 4 cm, how much soup thehospital has to prepare daily to serve 250 patients ?

387. Find the volume of the right circular cone with(i) radius 6 cm, height 7 cm (ii) radius 3.5 cm, height 12

cm388. The height of a cone is 15 cm. If its volume is 1570

cm3, find the radius of the base. (Use = 3.14)389. If the volume of a right circular cone of height 9 cm is

48 cm3, find the diameter of its base.390. The volume of a right circular cone is 9856 cm3. If the

diameter of the base is 28 cm, find(i) height of the cone (ii) slant height of the cone (iii)

curved surface area of the cone

391. A heap a wheat is in the form of a cone whosediameter is 1 0.5 m and height is 3 m. Find its volume.The heap is to e covered by canvas to protect it fromrain. Find the area of the canvas required.

392. Find the volume of a sphere whose radius is(i) 7 cm (ii) 0.63 m393. How many litres of milk can a hemispherical bowl of

diameter 10.5 cm hold ?394. Find the volume of a sphere whose surface area is 154

cm2.395. A capsule of medicine is in the shape of a sphere of

diameter 3.5 mm. How much medicine (inmm3) is needed to fill this capsule ?396. In quadrilateral ACBD, AC = AD and AB bisects ÐA.

Show that ABC ABD. What can you say aboutBC and BD?

397. ABCD is a quadrilateral in which AD = BC and ÐDAB =ÐCBA. Prove that

(i) DABD DBAC(ii) BD = AC(iii) ÐABD = ÐBAC


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398. Line is the bisector of an angle ÐA and B is anypoint on . BP and BQ are perpendicular from B to thearms of ÐA. Show that:

(i) DAPB DAQB(ii) BP = BQ or B is equidistant from the arms of ÐA.399. In DABC, AD is the perpendicular bisector of BC. Show

that DABC is an isosceles triangle in which AB = AC.

400. ABC is an isosceles triangle in which altitudes BE andCF are drawn to equal sides AC and AB respectively.Show that these altitudes are equal.

401. ABC is a right angled triangle in which ÐA = 900 andAB = AC. Find ÐB and ÐC.

402. AD is an altitude of an isosceles triangle ABC in whichAB = AC. Show that :

(i) AD bisects BC(i) AD bisects ÐA.403. BE and CF are two equal altitudes of a triangle ABC.

Using RHS congruence rule, prove that the triangleABC is isosceles.

404. ABC is an isosceles triangle with AB = AC. Draw APBC to show that ÐB = ÐC.

405. AB and CD are respectively the smallest and longestsides of a quadrilateral ABCD. Show that ÐA

> ÐC and ÐB > ÐD.

406. D is a point on side BC and DABC such that AD = AC.Show that AB > AD.

407. There is a slide in a park. One of its side walls hasbeen painted in some colour with a message “KEEP THEPARK GREEN AND CLEAN”. If the sides of the wall are 15m, 11 m and 6 m, find the area painted in colour.

408. Find the area of a triangle two sides of which are18cm and 10cm and the perimeter is 42 cm.

409. An isosceles triangle has perimeter 30cm and eachof the equal sides is 12cm. Find the area of the triangle.

410. A park, in the shape of a quadrilateral ABCD, has DC= 900, AB = 9m, BC = 12m, CD = 5m and AD = 8m. Howmuch area does it occupy ?

411. Find the area of a quadrilateral ABCD in which AB =3cm, BC = 4cm, CD = 4cm, DA = 5cm and AC = 5CM.

412. A rhombus shaped field has green grass for 18 cowsto graze. If each side of the rhombus is 30m and itslonger diagonal in 48m, how much area of grass field willeach cow be getting ?


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413. An umbrella is made by stitching 10 triangularpieces of cloth of two different colours, each piecemeasuring 20cm, 50 cm and 50 cm. How much cloth ofeach colour is required for the umbrella ?

414. A kite in the shape of square with a diagonal 32 cmand an isosceles triangle of base 8cm and sides 6cmeach is to be made of three different shades as shown infig. How much paper of each shade has been used in it ?

415. A plastic box 1.5 m long, 1.25 m wide and 65 cmdeep is to be made. It is opened at the top. Ignoring thethickness of the plastic sheet, determine :(i) The area of the sheet required for making the box.(ii) The cost of sheet for it, if a sheet measuring 1 m2costs Rs. 20.

416. The length, breadth and height of a room are 5 m, 4m and 3 m respectively. Find the cost of white washingthe walls of the room and the ceiling at the rate of Rs.7.50 per m2.

417. A cubical box has each edge 10 cm and anothercuboidal box is 12.5 cm long, 10 cm wide and 8 cm high.(i) Which ox has the greater lateral surface area and byhow much ?(ii) Which box has the smaller total surface area and byhow much ?

418. A small indoor greenhouse (herbarium) is madeentirely of glass panes (including base) held togetherwith tape. It is 30 cm long, 25 cm wide and 25 cm high.(i) What is the area of the glass ?(ii) How much of tape is needed for all the 12 edges ?

419. It is required to make a closed cylindrical tank ofheight 1 m and base diameter 140 cm from a metalsheet. How many square metres of the sheet arerequired for the same ?

420. A metal pipe is 77 cm long. The inner diameter of across section is 4 cm, the outer diameter being 4.4 cm.Find its :(i) inner curved surface area,(ii) outer curved surface area,(iii) total surface area.

421. A cylindrical pillar is 50 cm is diameter and 3.5 m inheight. Find the cost of painting the curved surface ofthe pillar at the rate of Rs. 12.50 per m2.

422. Curved surface area of a right circular cylinder is 4.4m2. If the radius of the base of the cylinder is 0.7 m, findits height.

423. The inner diameter of a circular well is 3.5 m. It is 10m deep. Find(i) its inner curved surface area,(ii) the cost of plastering this curved surface at the rateof Rs 40 per m2

424. Find the total surface area of a cone, if its slantheight is 21 m and diameter of its base is 24 m.

425. A joker’s cap is in the form of a right circular cone ofbase radius 7 cm and height 24 cm. Find the area of thesheet required to make 10 such caps.

426. A bus stop is barricaded from the remaining part ofthe road, by using 50 hollow cones made of recycledcardboard, Each cone has a base diameter of 40 cm andheight 1 m. If the outer side of each of the cones is to bepainted and the cost of painting is Rs 12 per m2, whatwill be the cost of painting all these cones ?

(Use = 3.14 and take = 1.02)427. Find the surface area of a sphere of radius :

(i) 10.5 cm (ii) 5.6 cm (iii) 14 cm428. A hemispherical bowl made of brass has inner

diameter 10.5 cm. Find the cost of tin-plating it on theinside at the rate of Rs. 16 per 100 cm2.

429. A cuboidal water tank is 6 m long, 5 m wide and 4.5m deep. How many litres of water can it hold? (1 m3 = 1000 )

430. Find the cost of digging a cuboidal pit 8 m long, 6 mbroad and 3 m deep at the rate of Rs. 30 per m3.

431. A godown measures 40 m × 25 m × 10 m. Find themaximum number of wooden crates each measuring 1.5m × 1.25 m × 0.5 m that can be stored in the godown.

432. A river 3 m deep and 40 m wide is flowing at therate of 2 km per hour. How much water will fall into thesea in a minute ?


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433. A soft drink is available in two packs - (i) a tin canwith a rectangular base of length 5 cm and width 4 cm,having a height of 15 cm and (ii) a plastic cylinder withcircular base of diameter 7 cm and height 10 cm. Whichcontainer has greater capacity and by how much ?

434. If the lateral surface of a cylinder is 94.2 cm2 and itsheight is 5 cm, then find(i) radius of its base (ii) its volume. (Use = 3.14)

435. A lead pencil consist of a cylinder of wood with asolid cylinder of graphite filled in the interior. Thediameter of the pencil is 7 mm and the diameter of thegraphite is 1 mm. If the length of the pencil is 14 cm,find the volume of the wood and that of the graphite.

436. A patient in a hospital is given soup daily in acylindrical bowl of diameter 7 cm. If the bowl is filledwith soup to a height of 4 cm, how much soup thehospital has to prepare daily to serve 250 patients ?

437. Find the volume of the right circular cone with(i) radius 6 cm, height 7 cm (ii) radius 3.5 cm, height 12cm

438. The height of a cone is 15 cm. If its volume is 1570cm3, find the radius of the base. (Use = 3.14)

439. If the volume of a right circular cone of height 9 cmis 48 cm3, find the diameter of its base.

440. The volume of a right circular cone is 9856 cm3. Ifthe diameter of the base is 28 cm, find(i) height of the cone (ii) slant height of the cone

(iii) curved surface area of the cone441. A heap a wheat is in the form of a cone whose

diameter is 1 0.5 m and height is 3 m. Find its volume.The heap is to e covered by canvas to protect it fromrain. Find the area of the canvas required.

442. Find the volume of a sphere whose radius is(i) 7 cm (ii) 0.63 m

443. How many litres of milk can a hemispherical bowl ofdiameter 10.5 cm hold ?

444. Find the volume of a sphere whose surface area is154 cm2.

445. A capsule of medicine is in the shape of a sphere ofdiameter 3.5 mm. How much medicine (inmm3) is needed to fill this capsule ?

446. In quadrilateral ACBD, AC = AD and AB bisects ÐA.Show that ABC ABD. What can you say aboutBC and BD?

447. ABCD is a quadrilateral in which AD = BC and ÐDAB= ÐCBA. Prove that(i) DABD DBAC(ii) BD = AC(iii) ÐABD = ÐBAC

448. Line is the bisector of an angle ÐA and B is anypoint on . BP and BQ are perpendicular from B to thearms of ÐA. Show that:

(i) DAPB DAQB(ii) BP = BQ or B is equidistant from the arms of ÐA.

449. In DABC, AD is the perpendicular bisector of BC.Show that DABC is an isosceles triangle in which AB =AC.


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450. ABC is an isosceles triangle in which altitudes BE andCF are drawn to equal sides AC and AB respectively.Show that these altitudes are equal.

451. ABC is a right angled triangle in which ÐA = 900 andAB = AC. Find ÐB and ÐC.

452. AD is an altitude of an isosceles triangle ABC inwhich AB = AC. Show that :(i) AD bisects BC(i) AD bisects ÐA.

453. BE and CF are two equal altitudes of a triangle ABC.Using RHS congruence rule, prove that the triangle ABCis isosceles.

454. ABC is an isosceles triangle with AB = AC. Draw APBC to show that ÐB = ÐC.

455. AB and CD are respectively the smallest and longestsides of a quadrilateral ABCD. Show that ÐA> ÐC and ÐB > ÐD.

456. D is a point on side BC and DABC such that AD = AC.Show that AB > AD.

457. Find the area of a triangle whose sides are 13cm, 14 cm and 15 cm

458. Find the area of a triangle, two sides of whichare 8 cm and 11 cm and 11 cm the perimeter is 32cm.

459. The perimeter of a triangular field is 450 m andits sides are in the ratio 13 : 12 : 5. Find the area oftriangle.

460. The lengths of the sides of a triangle are 5 cm,12 cm and 13 cm. Find the length of perpendicularfrom the opposite vertex to the side whose length is13 cm.

461. The triangular side walls of a flyover have beenused for advertisements. The sides of the walls are122 m, 22m and 120 m. The advertisements yieldan earning of Rs. 5000 per m2 per year. A companyhired both walls for 3 months. How much rent did itpay? [NCERT]

462. Find the area of the quadrilateral ABCD, inwhich AB = 7 cm, BC = 6 cm, CD = 12 cm, DA = 15cm and AC = 9 cm.

463. In fig. ABCD is a field in the form of aquadrilateral whose sides are indicated in the figure.If ÐDAB = 900, find the area of the field.


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464. Find the area of trapezium whose parallel sides25 cm, 13 cm and other sides are 15 cm and 15 cm.

465. Sanya has a piece of land which is in the shapeof a rhombus. She wants her one daughter and oneson to work on the land and produce different cropsto suffice the needs of their family. She divided theland in two equals parts. If the perimeter of the landis 400 m and one of the diagonals is 160 m, howmuch area each of them will get ?[NCERT]

466. Parveen wanted to make a temporary shelter forher car, by making a box-like structure withtarpaulin that covers all the four sides and the top ofthe car (with the front face as a flap which can berolled up). Assuming that the stitching margins arevery small, and therefore negligible, how muchtarpaulin would be required to make the shelter ofheight 2.5 m, with base dimensions 4 m × 3 m?

467. Find the surface area of a cube whose edge is15 cm.

468. The paint in a certain container is sufficient topaint an area equal to 9.375 m2. How many bricksof dimensions 22.5 cm × 10 cm × 7.5 cm can bepainted out of this container [NCERT]

469. A small indoor greenhouse is made entirely ofglass sheets (including the base) held together withtape. It is 40 cm long, 30 cm wide and 30 cm high.Find(i) the area of the glass sheet required and(ii) the total length of the tape required for all the 12edges.

470. A matchbox measures 4 cm × 2.5 cm × 1.5 cm.What will be the volume of a packet containing 12such boxes ?

471. A wall of length 10 m was to be built across anopen ground. The height of the wall is 4 m andthickness of the walls is 24 cm. If this wall is to bebuilt up with bricks of dimensions 24 cm × 12 cm ×8 cm, then find the number of bricks which arerequired.

472. Akriti playing with plastic building blocks whichare of identical cubical shapes. She makes astructure as shown in fig. If the edge of each cube is5 cm, then find the volume of the structure. built byAakriti.

473. In a hot water heating system, there is acylindrical pipe of length 28 m, and diameter 5 cm.Find the total radiating surface in the system.

474. A cylindrical block of wood has radius 70 cmand length 2 m is to be painted with blue colouredenamel. The cost of painting is Rs. 1.25 per 100cm2. Find the cost of painting the block. Take

.475. The curved surface area of a right circular

cylinder of height 14 cm is 88 cm2. Find thediameter of the base of the cylinder.

476. A cylindrical vessel, without lid, has to betin-coated including both of its sides. If the radius of

its base is m and its height is 1.4 m, calculatethe cost of tin-coating at the rate of Rs. 50 per 1000cm2 (Use = 3.14)


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477. The diameter of a roller is 84 cm and its lengthis 120 cm. It takes 500 complete revolutions tomove once over to level a playground. Find the areaof the playground in m2. [NCERT]

478. The pillars of a temple are cylindrical shaped. Ifeach pillar has a circular base of radius 20 cm, andheight 7m, then find the quantity of concrete mixtureused to build 20 such pillars. Also find the cost ofthe concrete mixture at the rate of Rs. 200 per m3.

479. The circumference of the base of a cylindricalvessel is 132 cm and its height is 25 cm. How manylitres of water can it hold? (1000 cm3 = 1 )

480. The inner diameter of a cylindrical wooden pipeis 24 cm and its outer diameter is 28 cm. The lengthof the pipe 35 cm. Find the mass of the pipe. If 1cm3 of wood has a mass of 0.6 g.

481. Diameter of the base of a cone is 10.5 cm andits slant height is 10 cm. Find its curved surfacearea.

482. The radius of the base of a conical tent is 12 m.The tent is 9 m high. Find the cost of the canvasrequired to make the tent, if one square metre ofcanvas costs Rs. 120. (Take = 3.14)

483. Curved surface area of a cone is 308 cm2 andits slant height is 14 cm. Find(i) radius of the base and (ii) total surface area ofthe cone.

484. How many metres of cloth of 1.1 m width will berequired to make conical tent whose vertical heightis 12 m and base radius is 16 m? Find also the costof the cloth used at the rate of Rs. 41 per metre.

485. A conical pit of top diameter 3.5 m is 12 mdeep. What is its capacity in kilolitres [NCERT]

486. The curbed surface of a right circular cone is198 cm2 and the radius of its base is 7 cm. Find the

volume of the cone.487. Find the capacity in litres of a conical vessel

with (i) radius 7 cm, slant height 25 cm. (ii) height12 cm, slant height 13 cm.

488. Find the surface area of a sphere of diameter 14m.

489. Find the radius of sphere whose surface area is314 cm3. (Use = 3.14)

490. Find the total surface area of a hemisphere ofradius 10 cm. (Use = 3.14)

491. A hemispherical bowl is made from a metalsheet having thickness 0.3 cm. The inner radius ofthe bowl is 24.7 cm. Find the cost of polishing itsouter surface at the rate of Rs. 4 per 100 cm2.(Take = 3.14)

492. Find the amount of water displaced by a solidspherical ball of diameter 28 cm. [NCERT]

493. There are 42 hemispherical bowls, each ofradius 3.5 cm. Find the quantity of water in litreswhich is just sufficient to fill these 42 bowls.

494. A hemispherical tank is made up of an ironsheet 1 cm thick. If the inner radius is 1 m, then findthe volume of the iron used to make the tank.

495. Find the volume of a sphere whose surface areais 55.44 cm2.

496. In fig. OA = OB and OD = OC. Show that(i) DAOD DBOC and (ii) AD || BC.

497. In DABC, AB = AC. If P is a point of AB and Q isa point on AC such that AP = AQ. Prove that(i) DAPC DAQB (ii) DBPC DCQB.

498. In figure, AC = AE, AB = AD and ÐBAD =ÐEAC, prove that BC = DE.

499. In figure, diagonal AC of a quadrilateral ABCDbisects he angles A and C. Prove that AB = AD andCB = CD.


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500. AB is a line segment and P is its mid-point. Dand E are points on the same side of AB such thatÐBAD = ÐABE and ÐEPA = ÐDPB.Show that (i) DDAP DEBP (ii) AD =BE.501. AB is a line-segment. AX and BY are two equalline-segments drawn on opposite sides of line ABsuch that AX || BY. If AB and XY intersect eachother at P. Prove that :(i) DAPX DBPY(ii) AB and XY bisect each otherat P.

502. In fig., AD is a median and BL, CM areperpendiculars drawn from B and C respectively onAD and AD produced. Prove that BL = CM.

503. AD and BC are equal perpendiculars to a linesegment AB. Show that CD bisects AB.

504. If fig. ÐBCD = ÐADC and ÐACB = ÐBDA.Prove that AD = BC and ÐA = ÐB.

505. In the fig. ABCD is a quadrilateral in which AB =AD and BC = DC. Prove that :(i) AC bisects each of the angles A and C.(ii) BE = ED.(iii) ÐABC = ÐADC. Can we say that AE = EC ?

506. AB is a line segment, P and Q are points onopposite sides of AB such that each of them isequidistant from the points A and B. Show that theline PQ is the perpendicular bisector of AB.

507. DABC and DPBC are two isosceles triangles onthe same base BC and vertices A and P are on thesame side of BC. A and P are joined, show that :(i) DABP DACP and (ii) AP bisects ÐA ofDABC.

508. In fig. P is a point equidistant from the linesand m intersecting at point A. Show that the line n(along AP) bisects the angle between and m.


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509. AD, BE and CF, the altitudes of DABC areequal. Prove that DABC is an equilateral triangle.

510. In a DABC, the internal bisectors of ÐB and ÐCmeet at O. Prove that OA is the internal bisector ofÐA.

511. In the adjoining fig, find the value of x.

512. DABC is an isosceles triangle in which AB = AC.Side BA is produced to D such that AD = AB. Showthat ÐBCD is a right angle.

513. In a right angled triangle, one acute angle isdouble the other. Prove that the hypotenuse isdouble the smallest side.

514. In fig, show that : (i) AB > AC (ii) AB > BC and(iii) BC > AC.

515. Show that of all the line segments that can bedrawn to a given straight line from a given pointoutside it, the perpendicular is the shortest.

516. Show that the sum of the three altitudes of atriangle is less than the sum of the three sides ofthe triangle.

517. The area of a triangle is 30 cm2. Find the base ifthe altitude exceeds the base by 7 cm.

518. The cost of turfing a triangle field at the rate ofRs. 45 per 100 m2 is Rs. 900. Find the height, ifdouble the base of the triangle is 5 times the height.

519. From a point in the interior of an equilateraltriangle, perpendicular drawn to the three sides are8 cm, 10 cm and 11 cm respectively. Find the area ofthe triangle.

520. The difference between the sides at right anglesin a right-angled triangle is 14 cm. The area of thetriangle is 120 cm2. Calculate the perimeter of thetriangle.

521. Find the percentage increase in the area of atriangle if its each side is doubled.

522. An umbrella is made by stitching 10 triangularpieces of cloth of two different colours (see figure),each piece measuring 20 cm, 50 cm and 50 cm. Howmuch cloth of each colour is required for theumbrella ?

523. Three equal cubes are placed adjacently in a row.Find the ratio of the total surface area of the new

cuboids to that of the sum of the surface areas ofthree cubes.524. A class room is 7 m long, 6.5 m wide and 4 m high. Ithas one door 3 m × 1.4 m and three windows each

measuring 2 m × 1 m The interior walls are to becolour-washed. The contractor charges Rs. 15 persq.m. Find the cost of colour washing.

525. A cylindrical vessel, without lid, has to be tin coatedincluding both of its sides. If the radius of its base

is m and its height is 1.4 m, calculate the cost oftin-coating at the rate of Rs. 50 per 1000 cm2.527. How many metres of cloth of 1.1 m width will berequired to make a conical tent whose vertical height

is 12 m and base radius is 16 m ? Find also the cost ofthe cloth used at the rate of Rs 14 per metre.528. The surface area of a sphere of radius 5 cm is fivetimes the area of the curved surface of cone of radius 4

cm. Find the height of the cone.


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529. The dimensions of a cinema hall are 100 m, 50 m and18m. How many persons can sit in the hall, if

each required 150 m3 of air ?

530. The outer measurements of a closed wooden box are42 cm, 30 m and 27 cm. If the box is made of 1 cm

thick wood, determine the capacity of the box.531. If v is the volume of a cuboids of dimensions a,b, andc and s is its surface area, then prove that

532. The ratio of the volumes of the two cones is 4 : 5 andthe ratio of the radii of their bases is 2 : 3. Find the

ratio of their vertical heights.533. If h, c and v be the height, curved surface and

volume of a cone, show that 3 - c2h2 + 9v2 = 0.534. How many balls, each of radius 1 cm, can be madefrom a solid sphere of lead of radius 8 cm ?535. By melting a solid cylindrical metal, a few conicalmaterials are to be made. If three times the radius of

the cone is equal to twice the radius of the cylinderan the ratio of the height of the cylinder and theheight of the cone is 4 : 3, find the number of coneswhich can be made.

536. Water flows at the rate of 10 per minute through acylindrical pipe having its diameter as 5 mm. How

much time will it take to fill a conical vessel whosediameter of the base is 40 cm and depth 24 cm ?537. The whole surface of a rectangular lock is 846 cm2.Find the length, breadth and height, if these

dimensions are in the ratio 5 : 4 : 3.538. An open box is made of wood 3 cm thick. its externallength, breadth and height are 1.48 m, 1.16 m and

8.3 dm. Find the cost of painting the inner surface atRs 5 per m2.539. A room 8 m long 6 m board and 3 m high has two

windows m × 1 m and a door 2 m × m. Findthe cost of papering the walls will paper 50 cm wide

at Rs. 40 per metre.540. 50 circular plates, each of radius 7 cm and thickness

cm, are placed one above the other to form a solid rightcircular cylinder. Find the total surface area.

541. A tent in the shape of a right circular cylindersurmounted by a right circular cone. The heights of the

cylindrical and the conical parts are 40 m and 21 mrespectively. If the base diameter of the tent is 56 m,find the area of the required canvas to make this tentif 20% of the area is consumed in folding and sewing.

542. A toy is in the form of a right circular cylinder closedat one end and with a hemisphere on the other

end. The height and the radius of the base are 15 cmand 6 cm respectively. The radius of the hemisphereare cylinder are same. Calculate the total surface areaand the volume of the toy. if the toy is painted at therate of Rs. 2.50 per 10 cm2, find the cost of paintingthe toy.

543. An iron pillar has some portion in the form of a rightcircular cylinder an remaining in the form of a

right circular cone. The radius of the base of each ofthe cone and the cylinder is 8 cm. The cylindricalportion is 240 cm high and the conical part is 36 cmhigh. Find the weight of the pillar, if one cubic cm ofiron weights 7.8 g.

544. A solid metallic sphere of diameter 28 is melted andrecasted into a number of smaller cones, each of

diameter cm and height 3 cm. Find the numberof cones so formed.


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5) surface areas & volumes - questions

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