Mensuration

  1. A pit is dug in the shape of a cuboid with dimensions 10m x 8m x 3m. The earth taken out is spread evenly on a rectangular plot of land with dimensions 40 m x 30 m. What is the increase in the level of the plot?
  2. The area of cross section of a pipe is 5.4cm2 and water is pumped out of it at the rate of 27km/h. Find in liters the volume of water which flows out of the pipe in one minute.
  3. 160 m3 of water is to be used to irrigate a rectangular field whose area is 800 m2. What will be the height of the water level in the field?
  4. The parallel sides of a trapezium are 40 cm and 20 cm. If its non-parallel sides are both equal, each being 26 cm, find the area of the trapezium.
  5. Find the area of polygon ABCDEF, if AD = 18cm, AQ = 14 cm, AP = 12 cm, AN = 8 cm, AM = 4 cm, and FM, EP, QC and BN are perpendiculars to diagonal AD.
  6. Horse stable is in the form of a cuboid, whose external dimensions are 70 m × 35 m × 40 m, surrounded by a cylinder halved vertically through diameter 35 m and it is open from one rectangular face 70 m × 40 m. Find the cost of painting the exterior of the stable at the rate of Rs 2/m2.
  7. A cube of side 5 cm is painted on all its faces. If it is sliced into 1 cubic centimetre cubes, how many 1 cubic centimetre cubes will have exactly one of their faces painted?
  8. A cube of side 4 cm is cut into 1 cm cubes. What is the ratio of the surface areas of the original cubes and cut-out cubes?
  9. A circle of maximum possible size is cut from a square sheet of board. Subsequently, a square of maximum possible size is cut from the resultant circle. What will be the area of the final square?
  10. What is the area of the largest triangle that can be fitted into a rectangle of length l units and width w units?
  11. The surface area of the three co-terminus faces of a cuboid are 6, 15 and 10 cm2 respectively. Find the volume of the cuboid.
  12. A regular hexagon is inscribed in a circle of radius r. Find the perimeter of the regular hexagon.
  13. How many small cubes with edge of 20 cm each can be just accommodated in a cubical box of 2 m edge?
  14. The volume of a cube is 64 cm3. Find its surface area.
  15. A bicycle wheel makes 500 revolutions in moving 1 km. Find the diameter of the wheel.
  16. A cube of side 5 cm is cut into as many 1 cm cubes as possible. What is the ratio of the surface area of the original cube to that of the sum of the surface areas of the smaller cubes?
  17. A square sheet of paper is converted into a cylinder by rolling it along its side. What is the ratio of the base radius to the side of the square?
  18. A wooden box (including the lid) has external dimensions 40 cm by 34 cm by 30 cm. If the wood is 1 cm thick, how many cm3 of wood is used in it?
  19. The internal dimensions of a rectangular room are 6 m, 5 m, and 3.5 m. It has two doors of size 1.2 m by 2 m and three windows of size 1 m by 1.9 m. The walls of the room are to be papered with a wallpaper of width 70 cm. Find the cost of the paper at the rate of 6.50 per meter.
  20. A river 2 m deep and 45 m wide is flowing at the rate of 3 km per hour. Find the amount of water in cubic metres that runs into the sea per minute.
  21. A truck carrying 7.8 m3 concrete arrives at a job site. A platform of width 5 m and height 2 m is being constructed at the site. Find the length of the platform, constructed from the amount of concrete on the truck?
  22. A swimming pool is 200 m by 50 m and has an average depth of 2 m. By the end of a summer day, the water level drops by 2 cm. How many cubic metres of water is lost on the day?
  23. A housing society consisting of 5,500 people needs 100 L of water per person per day. The cylindrical supply tank is 7 m high and has diameter 10 m. For how many days will the water in the tank last for the society?
  24. Metallic discs of radius 0.75 cm and thickness 0.2 cm are melted to obtain 508.68 cm3 of metal. Find the number of discs melted (use ­ = 3.14).
  25. The ratio of the radius and height of a cylinder is 2:3. If its volume is 12,936 cm3, find the total surface area of the cylinder.
  26. External dimensions of a closed wooden box are in the ratio 5:4:3. If the cost of painting its outer surface at the rate of Rs 5 per dm2 is Rs 11,750, find the dimensions of the box.
  27. The capacity of a closed cylindrical vessel of height 1 m is 15.4 L. How many square metres of metal sheet would be needed to make it?
  28. A rectangular sheet of dimensions 25 cm × 7 cm is rotated about its longer side. Find the volume and the whole surface area of the solid thus generated.
  29. From a pipe of inner radius 0.75 cm, water flows at the rate of 7 m per second. Find the volume in litres of water delivered by the pipe in 1 hour.
  30. Four times the area of the curved surface of a cylinder is equal to 6 times the sum of the areas of its bases. If its height is 12 cm, find its curved surface area.
  31. A cylindrical tank has a radius of 154 cm. It is filled with water to a height of 3 m. If water to a height of 4.5 m is poured into it, what will be the increase in the volume of water in kl?
  32. The length, breadth and height of a cuboidal reservoir is 7 m, 6 m and 15 m respectively. 8400 L of water is pumped out from the reservoir. Find the fall in the water level in the reservoir.
  33. How many bricks of size 22 cm × 10 cm × 7 cm are required to construct a wall 11m long, 3.5 m high and 40 cm thick, if the cement and sand used in the construction occupy (1/10)th part of the wall?
  34. Ratio of area of a circle to the area of a square whose side equals radius of circle is ______________ ­.
  35. A rectangular examination hall having seats for 500 candidates has to be built so as to allow 4 cubic metres of air and 0.5 square metres of floor area per candidate. If the length of hall be 25 m, find the height and breadth of the hall.
  36. The ratio between the curved surface area and the total surface area of a right circular cylinder is 1:2. Find the ratio between the height and radius of the cylinder.
  37. Water flows from a tank with a rectangular base measuring 80 cm by 70 cm into another tank with a square base of side 60 cm. If the water in the first tank is 45 cm deep, how deep will it be in the second tank?
  38. A rectangular sheet of paper is rolled in two different ways to form two different cylinders. Find the volume of cylinders in each case if the sheet measures 44 cm × 33 cm.
  39. A pit is dug in the shape of a cuboid with dimensions 12m x 9m x 4m. The earth taken out is spread evenly on a rectangular plot of land with dimensions 45 m x 36 m. What is the increase in the level of the plot?
  40. The area of cross-section of a pipe is 6.5 cm², and water is pumped out at the rate of 25 km/h. Find in liters the volume of water that flows out of the pipe in one minute.
  41. 200 m³ of water is to be used to irrigate a rectangular field with an area of 1000 m². What will be the height of the water level in the field?
  42. The parallel sides of a trapezium are 50 cm and 30 cm. If its non-parallel sides are both equal, each being 40 cm, find the area of the trapezium.
  43. Find the area of polygon ABCDEF if AD = 20 cm, AQ = 16 cm, AP = 14 cm, AN = 10 cm, AM = 5 cm, and FM, EP, QC, and BN are perpendiculars to diagonal AD.
  44. A horse stable is in the form of a cuboid with external dimensions 80 m × 40 m × 50 m, surrounded by a cylinder halved vertically through diameter 40 m. It is open from one rectangular face of 80 m × 50 m. Find the cost of painting the exterior of the stable at the rate of Rs 3/m².
  45. A cube of side 6 cm is painted on all its faces. If it is sliced into 1 cubic centimeter cubes, how many 1 cubic centimeter cubes will have exactly two of their faces painted?
  46. A cube of side 5 cm is cut into 1 cm cubes. What is the ratio of the surface areas of the original cube and the cut-out cubes?
  47. A circle of the maximum possible size is cut from a square sheet of board. Subsequently, a square of the maximum possible size is cut from the resultant circle. What will be the area of the final square?
  48. What is the area of the largest triangle that can be fitted into a rectangle of length 10 units and width 6 units?
  49. The surface area of the three coterminous faces of a cuboid are 12, 20, and 30 cm² respectively. Find the volume of the cuboid.
  50. A regular hexagon is inscribed in a circle of radius r. Find the area of the regular hexagon.
  51. How many small cubes with an edge of 25 cm each can be just accommodated in a cubical box of 2.5 m edge?
  52. The volume of a cube is 125 cm³. Find its surface area.
  53. A bicycle wheel makes 600 revolutions in moving 1 km. Find the radius of the wheel.
  54. A cube of side 6 cm is cut into as many 1 cm cubes as possible. What is the ratio of the surface area of the original cube to that of the sum of the surface areas of the smaller cubes?
  55. A square sheet of paper is converted into a cylinder by rolling it along its side. What is the ratio of the base radius to the side of the square?
  56. A wooden box (including the lid) has external dimensions of 45 cm by 40 cm by 35 cm. If the wood is 2 cm thick, how many cm³ of wood is used in it?
  57. The internal dimensions of a rectangular room are 8 m, 6 m, and 4 m. It has two doors of size 1.5 m by 2.2 m and three windows of size 1.2 m by 2 m. The walls of the room are to be papered with wallpaper of width 75 cm. Find the cost of the paper at the rate of Rs 7 per meter.
  58. A river 3 m deep and 50 m wide is flowing at the rate of 4 km per hour. Find the amount of water in cubic meters that runs into the sea per minute.
  59. A truck carrying 9.6 m³ concrete arrives at a job site. A platform of width 6 m and height 2.5 m is being constructed at the site. Find the length of the platform constructed from the amount of concrete on the truck.
  60. A swimming pool is 150 m by 60 m and has an average depth of 3 m. By the end of a summer day, the water level drops by 3 cm. How many cubic meters of water is lost on the day?
  61. A housing society consisting of 6,000 people needs 120 L of water per person per day. The cylindrical supply tank is 8 m high and has a diameter of 12 m. For how many days will the water in the tank last for the society?
  62. Metallic discs of radius 0.8 cm and thickness 0.25 cm are melted to obtain 640 cm³ of metal. Find the number of discs melted (use π = 3.14).
  63. The ratio of the radius and height of a cylinder is 3:4. If its volume is 15,708 cm³, find the total surface area of the cylinder.
  64. External dimensions of a closed wooden box are in the ratio 6:5:4. If the cost of painting its outer surface at the rate of Rs 6 per dm² is Rs 13,500, find the dimensions of the box.
  65. The capacity of a closed cylindrical vessel of height 1.5 m is 18.84 L. How many square meters of metal sheet would be needed to make it?
  66. A rectangular sheet of dimensions 30 cm × 8 cm is rotated about its longer side. Find the volume and the whole surface area of the solid thus generated.
  67. From a pipe of inner radius 0.85 cm, water flows at the rate of 8 m per second. Find the volume in liters of water delivered by the pipe in 1 hour.
  68. Radius of a cylinder is r and the height is h. Find the change in the volume if the (a) height is doubled. (b) height is doubled and the radius is halved. (c) height remains same and the radius is halved.
  69. Radius of a cylinder is r and height is h. Find the change in the volume if: (a) the height is doubled. (b) the height is doubled and the radius is halved. (c) the height remains the same and the radius is halved.
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