NCERT Solutions for Chapter 13 Surface Areas and Volumes Class 9 Maths

Chapter Name

NCERT Solutions for Chapter 13 Surface Areas and Volumes

Class

Class 9

Topics Covered

  • Solid Shapes
  • Surface Area and Volume of Cube and Cuboid
  • Surface Area and Volume of Cylinder and Cone
  • Surface Area and Volume of Sphere and Hemisphere

Related Study Materials

  • NCERT Solutions for Class 9 Maths
  • NCERT Solutions for Class 9
  • Revision Notes for Chapter 13 Surface Areas and Volumes Class 9 Maths
  • Important Questions for Chapter 13 Surface Areas and Volumes Class 9 Maths
  • MCQ for for Chapter 13 Surface Areas and Volumes Class 9 Maths

Short Revision for Ch 13 Surface Areas and Volumes Class 9 Maths

  1. Cube, cuboid, cylinder, sphere, cone, hemisphere, frustum of a cone etc. are the solid figures. These are three dimensional figures. 
  2. Surface is a plane figure. 
  3. Surface area or total surface area of a cube = 6a2 , a represents side of a cube.
  4. Lateral surface area of a cube = 4a2 .
  5. Perimeter of a cube = 12a.
  6. Volume of a cube = a3 .
  7. Length of the diagonal of a cube = a√3.
  8. Surface area of a cuboid = 2(lb + bh + lh), l, b, h are the length, breadth and height of a cuboid respectively.
  9. Lateral surface area of a cuboid = 2(l + b) × h.
  10. Volume of a cuboid = l × b × h.
  11. Length of the diagonal of a cuboid = √(l2 + b2 + h2).
  12. Area of the four walls of a room = 2(l+b) × h.
  13. The maximum length of a rod that can be put in a cube or a cuboid is equal to the length of its longest diagonal.
  14. Curved or lateral surface area of a cylinder = 2Ï€rh, where r and h represent the radius of the base and height of a cylinder respectively.
  15. Total surface area of a cylinder = 2Ï€r(r+h).
  16. Volume of a cylinder = πr2h.
  17. Surface area of each base of a cylinder = πr2.
  18. Surface area of each base of a hollow cylinder = Ï€(R2 – r2), where R and r are the base radii of outer and inner cylinder respectively.
  19. External curved (lateral) surface area of a hollow cylinder = 2Ï€Rh.
  20. Internal curved (lateral) surface area of hollow cylinder = 2Ï€rh.
  21. Curved surface are of hollow cylinder = 2Ï€h(R + r).
  22. Total surface area of a hollow cylinder = 2Ï€(R + r)(h + R – r).
  23. Volume of a hollow cylinder = Ï€h(R2 – r2).
  24. Curved or lateral surface area of a cone = πrl, where r and l are the base radius and slant height of a cone respectively.
  25. Total surface area of a cone = πr(l + r).
  26. Volume of a cone = 1/3 πr2h, where r is base radius and h is height of the cone.
  27. Base area of a cone = Ï€r2 .
  28. Volume of a cone = 1/3(base area) × height.
  29. Curved or total surface area of a sphere = 4Ï€r2, where r is radius of the sphere.
  30. Volume of a sphere = 4/3Ï€r3 .
  31. Curved surface area of a hemisphere = 2Ï€r2.
  32. Total surface area of hemisphere = 3Ï€r2 .
  33. Volume of a hemisphere = 2/3Ï€r3 .
  34. Volume of spherical shell = (4/3)Ï€(R3 – r3), where R and r are radii of outer and inner sphere respectively.

Exercise 13.1 

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 1m2 costs 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 ceiling at the rate of Rs 7.50 per m2.


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 m2 is Rs.15000, find the height of the hall.
[Hint: Area of the four walls = Lateral surface area.]


4. The paint in a certain container is sufficient to paint an area equal to 9.375 m2. 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 another cuboidal box is 12.5cm long, 10 cm wide and 8 cm high
(i) Which box has the greater lateral surface area and by how much?
(ii) Which box has the smaller total surface area and by how much?


6. A small indoor greenhouse (herbarium) is made entirely of glass panes (including base) held together with tape. It is 30cm 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?


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×20cm×5cm and the smaller of dimension 15cm×12cm×5cm. For all the overlaps, 5% of the total surface area is required extra. If the cost of the cardboard is Rs. 4 for 1000 cm2, find the cost of cardboard required for supplying 250 boxes of each kind.


8. Praveen 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 2.5m, with base dimensions 4m×3m?

Solution

Total area of tarpaulin required = 2(l + b)× h + lb
= [2(4 + 3) × 2.5 + 4 ×3] m2 .

Exercise 13.2 

1. The curved surface area of a right circular cylinder of height 14 cm is 88 cm2. Find the diameter of the base of the cylinder. (Assume π =22/7 )


2. It is required to make a closed cylindrical tank of height 1m and base diameter 140cm from a metal sheet. How many square meters of the sheet are required for the same? Assume π = 22/7


3. A metal pipe is 77 cm long. The inner diameter of a cross section is 4 cm, the outer diameter being 4.4cm. (see fig. 13.11). Find its


(i) inner curved surface area,
(ii) outer curved surface area
(iii) total surface area
(Assume π=22/7)


4. 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 m2? (Assume π = 22/7)


5. A cylindrical pillar is 50 cm in diameter and 3.5 m in height. Find the cost of painting the curved surface of the pillar at the rate of Rs. 12.50 per m2.
(Assume π = 22/7)


6. Curved surface area of a right circular cylinder is 4.4 m2. If the radius of the base of the base of the cylinder is 0.7 m, find its height. (Assume π = 22/7)

Solution

2Ï€rh = 4.4

⇒ 2× 22/7 ×0.7 × h = 4.4

⇒ h = 1 m.


7. The inner diameter of a circular well is 3.5m. It is 10m deep. Find
(i) its inner curved surface area,
(ii) the cost of plastering this curved surface at the rate of Rs. 40 per m2.
(Assume π = 22/7)


8. In a hot water heating system, there is cylindrical pipe of length 28 m and diameter 5 cm. Find
the total radiating surface in the system. (Assume π = 22/7).


9. Find
(i) the lateral or curved surface area of a closed cylindrical petrol storage tank that is 4.2 m in diameter and 4.5m high.
(ii) How much steel was actually used, if 1/12 of the steel actually used was wasted in making the tank. (Assume π = 22/7)


10. In fig. 13.12, 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. (Assume Ï€ = 22/7) 



11. The students of 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? (Assume π =22/7)


Exercise 13.3 

1. Diameter of the base of a cone is 10.5 cm and its slant height is 10 cm. Find its curved surface area. (Assume π=22/7)


2. Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m. (Assume π = 22/7)


3. Curved surface area of a cone is 308 cm2 and its slant height is 14 cm. Find
(i) radius of the base and (ii) total surface area of the cone.
(Assume π = 22/7)


4. 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 1 m2 canvas is Rs 70.
(Assume π=22/7)


5. What length of tarpaulin 3 m wide will be required to make conical tent of height 8 m and base radius 6m? 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]


6. The slant height and base diameter of conical tomb are 25m and 14 m respectively. Find the cost of white-washing its curved surface at the rate of Rs. 210 per 100 m2. (Assume π = 22/7)


7. A joker’s cap is in the form of right circular cone of base radius 7 cm and height 24cm. Find the area of the sheet required to make 10 such caps. (Assume Ï€ =22/7)


8. 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 m2, what will be the cost of painting all these cones? (Use Ï€ = 3.14 and take √(1.04) =1.02)


Exercise 13.4 

1. Find the surface area of a sphere of radius:
(i) 10.5cm (ii) 5.6cm (iii) 14cm
(Assume π=22/7)


2. Find the surface area of a sphere of diameter:
(i) 14cm (ii) 21cm (iii) 3.5cm
(Assume π = 22/7

Solution

(i) Diameter (d) = 14 cm 
∴ Radius(r) = 14/2 = 7 cm 
Now, surface area of the sphere = 4Ï€r2 
= 4 × 22/7 × 7 × 7
= 4 × 22 × 7 
= 616 cm2 .


3. Find the total surface area of a hemisphere of radius 10 cm. [Use π=3.14]


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


5. A hemispherical bowl made of brass has inner diameter 10.5cm. Find the cost of tin-plating it on the inside at the rate of Rs 16 per 100 cm2. (Assume π = 22/7)

Solution

Inner diameter = 10.5 cm ⇒ Inner radius = 5.25cm. 


6. Find the radius of a sphere whose surface area is 154 cm2. (Assume π = 22/7)


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


8. A hemispherical bowl is made of steel, 0.25 cm thick. The inner radius of the bowl is 5cm. Find the outer curved surface of the bowl. (Assume π =22/7)

Sol. Inner radius = 5 cm.
∴ Outer radius = ( 5 + 0.25) cm  = 5.25 cm. 
∴ Outer curved surface area = 2 × 22/7 × (5.25)2 .
= 173.25 cm2 .

9. A right circular cylinder just encloses a sphere of radius r (see fig. 13.22). Find
(i) surface area of the sphere,
(ii) curved surface area of the cylinder,
(iii) ratio of the areas obtained in(i) and (ii).



Exercise 13.5 

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

Solution

Volume of the packet = 12 × volume of one match box 
= 12 × 4 × 2.5 × 1.5 cm3
= 180 cm3 


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

Solution

Volume of water tank  = 6 × 5 × 4.5 = 135 m3 .
∴ Water it can hold = 135 × 1000 l = 135000 l.


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

Solution

Let height of the vessel be h metres. 
Volume of the vessel = 380 m3 .
⇒ 10 × 8 × h = 380
⇒ h = 4.75 m. 


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

Solution

Volume of the cuboidal pit = 8 × 6 ×3 m3 .= 144 m3 .
Cost of digging the pit = ₹ 30 × 144 = ₹ 4,320.


5. 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.

Sol. Let the breadth of the tank be b metres. 
Capacity of the tank = 50000 l. 
Volume of the tank = (50000/1000)m3 =  50 m3 . 
∴ 2.5 × b × 10 = 50
⇒ b = 2 m. 


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


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


8. 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.


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


Exercise 13.6 

1. The circumference of the base of cylindrical vessel is 132cm and its height is 25cm.
How many litres of water can it hold? (1000 cm3= 1L) (Assume π = 22/7)


2. The inner diameter of a cylindrical wooden pipe is 24cm and its outer diameter is 28 cm. The length of the pipe is 35cm.Find the mass of the pipe, if 1cm3 of wood has a mass of 0.6g. (Assume Ï€ = 22/7)


3. A soft drink is available in two packs –
(i) a tin can with a rectangular base of length 5cm and width 4cm, having a height of 15 cm and (ii) a plastic cylinder with circular base of diameter 7cm and height 10cm. Which container has greater capacity and by how much? (Assume π =22/7)


4. If the lateral surface of a cylinder is 94.2cm2 and its height is 5cm, then find
(i) radius of its base  (ii) its volume.[Use Ï€ = 3.14]


5. It costs Rs 2200 to paint the inner curved surface of a cylindrical vessel 10m deep. If the cost of painting is at the 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
(Assume π = 22/7)


6. The capacity of a closed cylindrical vessel of height 1m is15.4 liters. How many square meters of metal sheet would be needed to make it? (Assume π = 22/7)



7. A lead pencil consists of a cylinder of wood with solid cylinder of graphite filled in the interior. The diameter of the pencil is 7 mm and the diameter of the graphite is 1 mm. If the length of the pencil is 14 cm, find the volume of the wood and that of the graphite. (Assume π = 22/7)

8. A patient in a hospital is given soup daily in a cylindrical bowl of diameter 7cm. If the bowl is filled with soup to a height of 4cm, how much soup the hospital has to prepare daily to serve 250 patients? (Assume π = 22/7)


Exercise 13.7 

1. Find the volume of the right circular cone with
(i) radius 6cm, height 7 cm
(ii) radius 3.5 cm, height 12 cm
(Assume π = 22/7)


2. Find the capacity in litres of a conical vessel with
(i) radius 7cm, slant height 25 cm (ii) height 12 cm, slant height 12 cm
(Assume π = 22/7)


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


4. If the volume of a right circular cone of height 9cm is 48Ï€cm3, find the diameter of its base.


5. A conical pit of top diameter 3.5m is 12m deep. What is its capacity in kiloliters?
(Assume π = 22/7)


6. The volume of a right circular cone is 9856cm3. If the diameter of the base is 28cm, find
(i) height of the cone
(ii) slant height of the cone
(iii) curved surface area of the cone
(Assume π = 22/7)


7. A right triangle ABC with sides 5cm, 12cm and 13cm is revolved about the side 12 cm. Find the volume of the solid so obtained.



8. If the triangle ABC in the Question 7 is revolved about the side 5cm, then find the volume of the solids so obtained. Find also the ratio of the volumes of the two solids obtained in Questions 7 and 8.



9. A heap of wheat is in the form of a cone whose diameter is 10.5 m and height is 3 m. Find its volume. The heap is to be covered by canvas to protect it from rain. Find the area of the canvas.
(Assume π = 22/7)


Exercise 13.8 

1. Find the volume of a sphere whose radius is
(i) 7 cm (ii) 0.63 m
(Assume π =22/7)


2. Find the amount of water displaced by a solid spherical ball of diameter
(i) 28 cm (ii) 0.21 m
(Assume π =22/7)



3.The diameter of a metallic ball is 4.2cm. What is the mass of the ball, if the density of the metal is 8.9 g per cm3? (Assume π=22/7)


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


5. How many litres of milk can a hemispherical bowl of diameter 10.5cm hold? (Assume π = 22/7)


6. A hemi spherical tank is made up of an iron sheet 1cm thick. If the inner radius is 1 m, then find the volume of the iron used to make the tank. (Assume π = 22/7)


7. Find the volume of a sphere whose surface area is 154 cm2. (Assume π = 22/7)


8. A dome of a building is in the form of a hemi sphere. From inside, it was white-washed at the cost of Rs. 4989.60. If the cost of white-washing isRs20 per square meter, find the
(i) inside surface area of the dome (ii) volume of the air inside the dome
(Assume π = 22/7)


9. 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 Sand S’.


10. A capsule of medicine is in the shape of a sphere of diameter 3.5mm. How much medicine (in mm3) is needed to fill this capsule? (Assume π = 22/7)


Exercise 13.9 

1. A wooden bookshelf has external dimensions as follows: Height = 110cm, Depth = 25cm,
Breadth = 85cm (see fig. 13.31). The thickness of the plank is 5cm 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 cm2 and the rate of painting is 10 paise per cm2, find the total expenses required for polishing and painting the surface of the bookshelf.


2. The front compound wall of a house is decorated by wooden spheres of diameter 21 cm, placed on small supports as shown in fig. 13.32. Eight such spheres are used forth is purpose, and are to be painted silver. Each support is a cylinder of radius 1.5cm and height 7cm and is to be painted black. Find the cost of paint required if silver paint costs 25 paise per cm2 and black paint costs 5 paise per cm2.



3. The diameter of a sphere is decreased by 25%. By what percent does its curved surface area decrease?

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