13. Surface Areas and Volumes

NCERT

13. Surface Areas and Volumes

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Important Note :

Surface area :

(i) The Surface Area of a Cube  .

(ii) The lateral surface area of a cube

(iii) The surface area of a cuboid

(iv) The lateral surface area of a cuboid  .

(v)Curved surface area of a cylinder

(vi) Total surface area of a cylinder  

(vii) Curved surface area of a cone  

(viii) Total surface area of a cone

(ix) Surface Area of a sphere  

(x) Curved surface area of a Hemisphere

(xi) Total surface area of a hemisphere

2. The Volume :

(i)  The volume of cube

(ii) The volume of a cuboid

(iii) The volume of a cylinder

(iv) The volume of cone

(v) The volume of a sphere

(vi) The volume of hemisphere .

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EXERCISE 13.1

Solution : Here ,  ,   and

(i) The area of the sheet required for making the box

(ii) The cost of sheet

 

Solution : Here,  m , m and  m 

 The area of the room  the lateral surface area + Area of ceiling

 

 

 

  

The cost of white washing the walls of the room and the ceiling

Solution :  Let  ,  and  be the length, breadth and height of the rectangular hall respectively .

     

 m

Area of the four walls

          

The cost of painting the four walls

A/Q ,

 m

Therefore , the height of the hall 8 m .

Solution:  Here ,  cm , cm and  cm

  The area of the brick

The number of brick 

Solution:  (i)  Cubical box :  Here ,   

The lateral surface area of cubical box

For cuboidal box : Here ,   ,   and     

The lateral surface area of cuboidal box

Therefore , the lateral surface area of cubical box is greater by

(ii)   Cubical box :  Here ,  

The surface area of cubical box

    

For cuboidal box : Here ,   ,  and   

The surface area of cuboidal box

 Total surface area of cuboidal box is greater by

Solution : (i) Here ,   , and   

The surface area of the small indoor greenhouse

 

(ii)  The sum of all edges of the glass

Solution :  For bigger box : Here ,   ,  and

   Area of bigger box

 

For smaller box : Here ,   ,  and

Area of bigger box

Area of two box

 

Area of overlap parts

The total surface area of the box  

 The cost of the one box (each kind of boxes)

The cost of  250 boxes

Solution: Here,  ,  and   

The area of shelter for car by making a box-like structure with tarpaulin

EXERCISE 13.2

Solution : let  be the radius of the right circular cylinder .

Here ,  

A/Q , 

So,   

Therefore , the diameter of the base of the cylinder is 2 cm .

Solution : Here,  ,  

and

The total curve surface area of the cylindrical tank

 

Required the metal sheet is  .

Solution : Here , , and

(i) The inner curved surface area

(ii) The outer curved surface area

(iii) The total surface area  Inner CSA  Outer CSA  Area of two base

 

 

 

Solution : Here,  , and  

The curve surface area of one revolution by the roller

 

The area of 500 complete revolutions  

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 ` 12.50 per .
6. Curved surface area of a right circular cylinder is 4.4 m2. If the radius of the base of the cylinder is 0.7 m, find its height.
7. 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 cost of plastering this curved surface at the rate of ` 40 per m².
8. In a hot water heating system, there is a cylindrical pipe of length 28 m and diameter 5 cm. Find the total radiating surface in the system.
9. Find
(i) the lateral or curved surface area of a closed cylindrical petrol storage tank that is 4.2 m in diameter and 4.5 m high.
(ii) how much steel was actually used, if  of the steel actually used was wasted in making the tank.
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.

11. 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 13.3

Assume , unless stated otherwise.
1. Diameter of the base of a cone is 10.5 cm and its slant height is 10 cm. Find its curved surface area.
2. Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m.
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.
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 .
5. 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).
6. The slant height and base diameter of a conical tomb are 25 m and 14 m respectively. Find the cost of white-washing its curved surface at the rate of Rs 210 per 100 m² .
7. 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.
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 π = 3.14 and take  )

EXERCISE 13.4

Assume   , unless stated otherwise.
1. Find the surface area of a sphere of radius:
(i) 10.5 cm (ii) 5.6 cm (iii) 14 cm
2. Find the surface area of a sphere of diameter:
(i) 14 cm (ii) 21 cm (iii) 3.5 m
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 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.
5. A hemispherical bowl made of brass has inner diameter 10.5 cm. Find the cost of tin-plating it on the inside at the rate of Rs 16 per 100 cm² .
6. Find the radius of a sphere whose surface area is 154 cm².
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 5 cm. Find the outer curved surface area of the bowl.
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 × 2.5 cm × 1.5 cm. What will be the volume of a packet containing 12 such boxes?
2. A cuboidal water tank is 6 m long, 5 m wide and 4.5 m deep. How many litres of water can it hold? (1 m³ = 1000 l)
3. A cuboidal vessel is 10 m long and 8 m wide. How high must it be made to hold 380 cubic metres of a liquid?
4. 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  .
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.
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 × 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.
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 3 m deep and 40 m wide is flowing at the rate of 2 km per hour. How much water will fall into the sea in a minute?

EXERCISE 13.6

Assume    , unless stated otherwise

1. 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? (1000  )
2. The inner diameter of a cylindrical wooden pipe is 24 cm and its outer diameter is 28 cm. The length of the pipe is 35 cm. Find the mass of the pipe, if 1 cm3 of wood has a mass of 0.6 g.
3. 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 a height of 15 cm and (ii) a plastic cylinder with circular base of diameter 7 cm and height 10 cm. Which container has greater capacity and by how much?
4. If the lateral surface of a cylinder is 94.2  and its height is 5 cm, then find (i) radius of its base (ii) its volume. (Use π = 3.14)

5. 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  , find (i) inner curved surface area of the vessel,
(ii) radius of the base,
(iii) capacity of the vessel.
6. The capacity of a closed cylindrical vessel of height 1 m is 15.4 litres. How many square metres of metal sheet would be needed to make it?
7. 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 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.
8. 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 13.7

Assume   , unless stated otherwise.
1. Find the volume of the right circular cone with
(i) radius 6 cm, height 7 cm (ii) radius 3.5 cm, height 12 cm
2. Find the capacity in litres of a conical vessel with (i) radius 7 cm, slant height 25 cm (ii) height 12 cm, slant height 13 cm
3. The height of a cone is 15 cm. If its volume is 1570  , find the radius of the base. (Use π = 3.14)
4. If the volume of a right circular cone of height 9 cm is  , find the diameter of its base.
5. A conical pit of top diameter 3.5 m is 12 m deep. What is its capacity in kilolitres?
6. The volume of a right circular cone is 9856  . 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
7. 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.
8. If the triangle ABC in the 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 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 required.

EXERCISE 13.8

Assume    , unless stated otherwise.
1. Find the volume of a sphere whose radius is
(i) 7 cm (ii) 0.63 m
2. Find the amount of water displaced by a solid spherical ball of diameter
(i) 28 cm (ii) 0.21 m
3. The diameter of a metallic ball is 4.2 cm. What is the mass of the ball, if the density of the metal is 8.9 g per  ?
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 10.5 cm hold?
6. 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.
7. Find the volume of a sphere whose surface area is 154  .
8. A dome of a building is in the form of a hemisphere. From inside, it was white-washed at the cost of Rs 4989.60. If the cost of white-washing is Rs 20 per square metre, find the
(i) inside surface area of the dome, (ii) volume of the air inside the dome.
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 S and S′.
10. A capsule of medicine is in the shape of a sphere of diameter 3.5 mm. How much medicine (in ) is needed to fill this capsule?

EXERCISE 13.9 (Optional)*

1. A wooden bookshelf has external dimensions as follows: Height = 110 cm, Depth = 25 cm , Breadth = 85 cm (see Fig. 13.31). The thickness of the plank 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   and the rate of painting is 10 paise per  , 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 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 the cost of paint required if silver paint costs 25 paise per  and black paint costs 5 paise per  .

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

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Posted 3 years ago