# Surface Areas and Volumes

**Exercise 1**

**1. ** A plastic box 1.5 m long, 1.25 m wide and 65 cm deep is to be made. It is to be open at the top. Ignoring the thickness 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 m^{2} cost Rs.20.

**2. ** The length, breadth and height of a room are 5 m, 4 m and 3 m respectively. Find the cost of white washing the walls of the room and the ceiling at the rate of Rs. 7.50 per m^{2}.

**3. ** The floor of a rectangular hall has a perimeter 250 m. If the cost of painting the four walls at the rate of Rs. 10 per m^{2} is Rs. 15000, find the height of the hall.

**4. ** The paint in a certain container is sufficient to paint an area equal to 9.375 m^{2}. How many bricks of dimensions 22.5 cm × 10 cm × 7.5 cm can be painted out of this container?

**5. ** A cubical box has each edge 10 cm and a cuboidal box is 10 cm wide, 12.5 cm long and 8 cm high.

(i) Which box has the greater lateral surface area and by how much?

(i) Which box has the smaller total surface area and how much?

**6. ** A small indoor green house (herbarium) is made entirely of glass panes (including base) held together with tape. It is 30 cm long, 25 cm wide and 25 cm high.

(i) What is the surface area of the glass?

(ii) How much of tape is needed for all the 12 edges?

**7.**Shanti Sweets Stall was placing an order for making cardboard boxes for packing their sweets. Two sizes of boxes were required. The bigger of dimensions 25 cm by 20 cm by 5 cm and the smaller of dimensions 15 cm by 12 cm by 5 cm. 5% of the total surface area is required extra, for all the overlaps. If the cost of the card board is Rs. 4 for 1000 cm

^{2}, find the cost of cardboard required for supplying 250 boxes of each kind.

**8.** Parveen wanted to make a temporary shelter for her car, by making a box-like structure with tarpaulin that covers all the four sides and the top of the car (with the front face as a flap which can be rolled up). Assuming that the stitching margins are very small and therefore negligible, how much tarpaulin would be required to make the shelter of height 2.5 m with base simensions 4 m × 3 m?

**Exercise 2**

**9.** The curved surface area of a right circular cylinder of height 14 cm is 88 cm^{2}. Find the diameter of the base of the cylinder.

**10. ** It is required to make a closed cylindrical tank of height 1 m and base diameter 140 cm from a metal sheet. How many square meters of the sheet are required for the same?

**11. ** A metal pipe is 77 cm long. The inner diameter of a cross section is 4 cm, the outer diameter being 4.4 cm. [See fig.]. Find its:

(i) Inner curved surface area

(ii) Outer curved surface area

(iii) Total surface area

**12.**The diameter of a roller is 84 cm and its length is 120 cm. It takes 500 complete revolutions to move once over to level a playground. Find the area of the playground in m

^{2}.

**13. ** A cylindrical pillar is 50 cm in diameter and 3.5 m in height. Find the cost of white washing the curved surface of the pillar at the rate of Rs. 12.50 per m^{2}.

**14. ** Curved surface area of a right circular cylinder is 4.4 m^{2}. If the radius of the base of the cylinder is 0.7 m, find its height.

**15. ** The inner diameter of a circular well is 3.5 m. It is 10 m deep. Find:

(i) Its inner curved surface area.

(ii) The cosr of plastering this curved surface at the rate of Rs. 40 per m^{2}.

**16.** In a hot water heating system, there is a cylindrical piping of length 28 m and diameter 5 cm. Find the total radiating surface in the system.

**17. ** Find:

(i) The lateral or curved surface area of a petrol storage tank that is 4.2 m in diameter and 5 m high.

(ii) How much steel was actually used if of the steel actually used was wasted in making 12 the tank?

**18.**In the adjoining figure, you see the frame of a lampshade. It is to be covered with a decorative cloth. The frame has a base diameter of 20 cm and height of 30 cm. A margin of 2.5 cm is to be given for folding it over the top and bottom of the frame. Find how much cloth is required for covering the lampshade. [See fig.]

**19.** The students of a Vidyalaya were asked to participate in a competition for making and decorating penholders in the shape of a cylinder with a base, using cardboard. Each penholder was to be of radius 3 cm and height 10.5 cm. The Vidyalaya was to supply the competitors with cardboard. If there were 35 competitors, how much cardboard was required to be bought for the competition?

**Exercise 3**

**20. **Diameter of the base of a cone is 10.5 cm and its slant height is 10 cm. Find its curved surface area and its total surface area.

**21.** Find the total surface area of a cone, if its slant height is 21 cm and diameter of the base is 24 cm.

**22. ** Curved surface area of a cone is 308 cm^{2} and its slant height is 14 cm. Find (i) radius of the base and (ii) total surface area of the cone.

**23. ** A conical tent is 10 m high and the radius of its base is 24 m. Find:

(i) Slant height of the tent.

(ii) Cost of the canvas required to make the tent, if the cost of a m^{2} canvas is Rs. 70.

**24. ** 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 be required for stitching margins and wastage in cutting is approximately 20 cm. Use π = 3.14)

**25.** The slant height and base diameter of a conical tomb are 25 m and 14 m respectively. Find the cost of whitewashing its curved surface at the rate of Rs. 210 per 100 m^{2}.

**26.** 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 of the sheet required to make 10 such caps.

**27.** A bus stop is barricaded from the remaining part of the road, by using 50 hollow cones made of recycled cardboard. Each cone has a base diameter of 40 cm and height 1 m. If the outer side of each of the cones is to be painted and the cost of painting is Rs. 12 per m^{2}, what will be the cost of painting all these cones? Use π* *= 3.14 and take √1.04 =1.02).

**Exercise 4**

**28. ** Find the surface area of a sphere of radius:

(i) 10.5 cm (ii) 5.6 cm (iii) 14 cm

**29. ** Find the surface area of a sphere of diameter:

(i) 14 cm (ii) 21 cm (iii) 3.5 cm

**30. ** Find the total surface area of a hemisphere of radius 10 cm. (Use π** = **3.14)

**31. ** The radius of a spherical balloon increases from 7 cm to 14 cm as air is being pumped into it. Find the ratio of surface areas of the balloon in the two cases.

**32.** A hemispherical bowl made of brass has inner diameter 105 cm. Find the cost of tin-plating it on the inside at the rate of Rs. 16 per 100 cm^{2}.

**33. ** Find the radius of a sphere whose surface area is 154 cm^{2}.

**34.** The diameter of the moon is approximately one fourth the diameter of the earth. Find the ratio of their surface areas.

**35.** A hemispherical bowl is made of steel, 0.25 cm thick. The inner radius of the bowl is 5 cm. Find the outer curved surface area of the bowl.

**36.** A right circular cylinder just encloses a sphere of radius *r *(See figure). Find:

(i) Surface area of the sphere.

(ii) Curved surface area of the cylinder.

(ii) Ratio of the areas obtained in (i) and (ii).

**Exercise 5**

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

**38. ** A cubical water tank is 6 m long, 5 m wide and 4.5 m deep. How many litres of water can it hold? (1 m^{3} = 1000 *l* )

**39. ** A cuboidal vessel is 10 m long and 8 m wide. How high must it be to hold 380 cubic meters of a liquid?

**40. ** 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 m^{3}.

**41.** The capacity of a cuboidal tank is 50000 litres of water. Find the breadth of the tank, if its length and depth are respectively 2.5 m and 10 m. (1 m^{3} = 1000 *l*)

**42. ** A village having a population of 4000 requires 150 litres of water per head per day. It has a tank measuring 20 m by 15 m by 6 m. For how many days will the water of this tank last?

**43.** A godown measures 40 m × 25 m × 15 m. Find the maximum number of wooden crates each measuring 1.5 m × 1.25 m × 0.5 m that can be stored in the godown.

**44. ** Find the minimum number of bricks each measuring 22.5 cm × 11.5 cm × 7.5 cm required to construct a wall 10 m long, 6 m high and 1.5 m thick.

**45. ** A solid cube of side 12 cm is cut into eight cubes of equal volume. What will be the side of the new cube? Also, find the ratio between their surface areas.

**46.** A river 3 m deep and 40 m wide is flowing at the rate of 2 km per hour. How much water eill fall into the sea in a minute?

**47. ** Find the length of a wooden plank of width 2.5 m, thickness 0.025 m and volume 0.25 m^{3}.

**Exercise 6**

**48.** The circumference of the base of a cylindrical vessel is 132 cm and its height is 25 cm. How many litres of water can it hold? (1 m^{3} = 1000 *l*)

**49. ** The inner diameter of a cylindrical wooden pipe is 24 cm and its out diameter is 28 cm. The length of the pipe is 35 cm. Find the mass of the pipe, if 1 cm^{3} of wood has a mass of 0.5 g.

**50.** A soft drink is available in two packs (i) a tin can with a rectangular base of length 5 cm and width 4 cm, having height of 15 cm (ii) a plastic cylinder with circular base of diameter 7 cm and height 10 cm. Which container has greater capacity and how much?

**51.** If the lateral surface of a cylinder is 94.2 cm^{2} and its height is 5 cm, then

(i) radius of its base (ii) volume of the cylinder.

**52. ** It costs Rs. 2200 to paint the inner curved surface of a cylindrical vessel 10 m deep. If the cost of painting is at the rate of Rs. 20 per m^{2}, find:

(i) Inner curved surface area of the vessel.

(ii) Radius of the base.

(iii) Capacity of the vessel.

**53. ** The capacity of a closed cylindrical vessel of height 1 m is 15.4 litres. How many square meters of metal sheet would be needed to make it?

**54. ** A bag of grain contains 2.8 m^{3} of grain. How many bags are needed to fill a drum of radius 4.2 m and height 5 m?

**55. ** A lead pencil consists of a cylinder of wood with a solid cylinder of graphite filled in the interior. The diameter of the pencil is 7 mm and diameter of graphite is 1 mm. If the length of the pencil is 14 cm, find the columns of the wood and that of the graphite.

**56.** A patient in a hospital is given soup daily in a cylindrical bowl of diameter 7 cm. If the bowl is filled with soup to a height of 4 cm, how much soup the hospital has to prepare daily to serve 250 patients?

**Exercise 7**

**57. ** Find the volume of the right circular cone with:

(i) Radius 6 cm, Height 7 cm

(ii) Radius 3.5 cm, Height 12 cm

**58.** Find the capacity of a conical vessel with:

(i) Radius 7 cm, Slant height 25 cm

(ii) Height 12 cm, Slant height 13 cm

**59.** The height of a cone is 15 cm. If its volume is 1570 cm^{3}, find the radius of the base. (Use π = 3.14)

**60. ** If the volume of a right circular cone of height 9 cm is 487t cm^{3}, find the diameter of the base.

**61. ** A conical pit of top diameter 3.5 m is 12 m deep. What is its capacity in kiloliters?

**62. ** The volume of a right circular cone is 9856 cm^{3}. If the diameter of the base if 28 cm, find:

(i) Height of the cone

(ii) Slant height of the cone

(iii) Curved surface area of the cone.

**63.**A right triangle ABC with sides 5 cm, 12 cm and 13 cm is revolved about the side 12 cm. Find the volume of the solid so obtained. (Use π = 3.14)

**64.** If the triangle ABC in question 7 above is revolved about the side 5 cm, then find the volume of the solid so obtained. Find, also, the ratio of the volume of the two solids obtained.

**65. ** Find the volume of the largest right circular cone that can be fitted in a cube whose edge is 14 cm.

**Exercise 8**

**66. ** Find the volume of a sphere whose radius is (i) 7 cm and (ii) 0.63 cm.

**67. ** Find the amount of water displaced by a solid spherical ball of diameter:

(i) 28 cm (ii) 0.21 m

**68. ** The diameter of a metallic ball is 4.2 cm. What is the mass of the ball, if the metal weighs 8.9 g per cm^{3}?

**69. ** The diameter of the moon is approximately one-fourth the diameter of the earth. What fraction is the volume of the moon of the volume of the earth?

**70. ** How many litres of milk can a hemispherical bowl of diameter 10.5 hold?

**71.** A hemispherical tank is made up of an iron sheet 1 cm thick. If the inner radius is 1 m, then find the volume of the iron used to make the tank.

**72.** Find the volume of a sphere whose surface area is 154 cm^{2}.

**73.** A dome of a building is in the form of a hemisphere. From inside, it was white-washed at the cost of Rs. 498.96. If the cost of white-washing is at the rate of Rs. 2.00 per square meter, find:

(i) The inner surface area of the dome.

(ii) The volume of the air inside the dome.

**74. ** Twenty seven solid iron spheres, each of radius *r *and surface area S are melted to form a sphere with surface area S’. Find the:

(i) Radius *r’ *of the new sphere.

(ii) Ratio of S and S’.

**75. ** A capsule of medicine is in the shape of a sphere of diameter 3.5 mm. How much medicine (in mm^{3}) is needed to fill this capsule?

**Exercise 9**

**76. ** A wooden bookshelf has external dimensions as follows: Height = 110 cm, Depth = 25 cm, Breadth = 85 cm [See fig.]. The thickness of the planks is 5 cm everywhere. The external faces are to be polished and the inner faces are to be painted. If the rate of polishing is 20 paise per cm^{2} and the rate of painting is 10 paise per cm2, find the total expenses required for polishing and painting the surface of the bookshelf.

**77.** The front compound wall of a house is decorated by wooden spheres of diameter 21 cm, placed on small supports as shown in figure. Eight such spheres are used for this purpose and are to be painted silver. Each support is a cylinder of radius 1.5 cm and height 7 cm and is to be painted black. Find he cost of paint required if silver paint costs 25 paise per cm^{2} and black paint costs 5 paise per cm^{2}.

**78.** If diameter of a sphere is decreased by 25% then what percent does its curved surface area decrease?

**79.** Sameera wants to celebrate the fifth birthday of her daughter with a party. She bought thick paper to make the conical party caps. Each cap is to have a base diameter of 10 cm and height 12 cm. A sheet of the paper is 25 cm by 40 cm and approximately 82% of the sheet can be effectively used for making the caps after cutting. What is the minimum number of sheets of paper that Sameera would need to buy, if there are to be 15 children at the party? (Use π = 3.14).