13. Surface Areas and Volumes

Class 9 Maths Chapter 13. Surface Areas and Volumes

Chapter 13. Surface Areas and Volumes

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 .

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 opened 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

costs Rs 20 .

Solution : Here , , and

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

(ii) The cost of sheet

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

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

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

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

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

Area of the four walls

The cost of painting the four walls

A/Q ,

Therefore , the height of the hall 8 m .

4. The paint in a certain container is sufficient to paint an area equal to 9.375

. How many bricks of dimensions 22.5 cm×10 cm×7.5 cm can be painted out of this container ?

Solution: Here , cm , cm and cm

The area of the brick

The number of brick

5. A cubical box has each edge 10 cm and another cuboidal box is 12.5 cm 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 ?

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

6. A small indoor greenhouse (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 area of the glass ? (ii) How much of tape is needed for all the 12 edges ?

Solution : (i) Here , , and

The surface area of the small indoor greenhouse

(ii) The sum of all edges of the glass

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 × 20 cm × 5 cm and the smaller of dimensions 15 cm ×12 cm ×5 cm . For all the overlaps, 5% of the total surface area is required extra . If the cost of the cardboard is Rs 4 for 1000

, find the cost of cardboard required for supplying 250 boxes of each kind .

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

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 dimensions

Solution: Here, , and

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

EXERCISE 13.2

Assume , unless stated otherwise.

1. The curved surface area of a right circular cylinder of height 14 cm is 88

. Find the diameter of the base of the cylinder.

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 .

2. 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 metres of the sheet are required for the same?

Solution : Here, ,

and

The total curve surface area of the cylindrical tank

Required the metal sheet is .

3. 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. 13.11). Find its

(i) inner curved surface area,
(ii) outer curved surface area,
(iii) total surface area.

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

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

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

Solution: Here, cm ,

and

The curve surface area of cylindrical pillar

Therefore, the cost of painting the curved surface of the pillar

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.

Solution: let be the height of cylinder .

Here,

A/Q,

Therefore, the height of cylinder is 1 m .

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 Rs 40 per m².

Solution: Here, ,

and

(i) The inner curved surface area of well

(ii) The cost of plastering of the well

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.

Solution: Here, , and

The curve surface area of the cylindrical pipe

Therefore, the total radiating surface in the system is 4.4 m² .

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.

Solution: Here, , and

(i) The lateral surface area of cylindrical petrol storage

(ii) let the actual area of steel used be .

The area of the wasted steel

A/Q,

Therefore, the actual area of steel used is 64.8 m² .

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.

Solution: Here, ,

and

The curve surface area of a lampshade

Therefore, the cloth is required for covering the lampshade is 2200 cm² .

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?

Solution: Here, and

The curve surface area of one the penholder

So, the curve surface area of 35 the penholder

Required the cardboard for the competition is 7920 cm² .

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.

Solution: Here, , and

The curve surface area of a cone

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

Solution: Here, , and

The total surface area of cone

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.

Solution: let be the radius of a cone .

Here,

A/Q ,

Therefore, the radius of a cone is 7 cm .

(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 .

Solution: Here, and

(i) The slant height of the tent

(ii) Here, ,and

The curve surface area of a canvas

The cost of the canvas required to make the tent

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).

Solution: Here, and

The slant height

The curve surface area of the conical tent

The length of tarpaulin

Therefore, the total length of tarpaulin

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 .

Solution: Here, and

The curve surface area of a conical tomb

Therefore , the cost of white-washing of a conical tomb

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.

Solution: Here, and

The slant height

The curve surface area of a joker’s caps

Therefore, the area of the sheet required to make 10 joker’s 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

)

Solution: Here, , and

The slant height

The curve surface area of a cone

The curve surface area of 50 cones

Therefore, the cost of painting of 50 cones

(approx.)

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

Solution: (i) Here,

The surface area of a sphere

(ii) Here,

The surface area of a sphere

(iii) Here,

The surface area of a sphere

2. Find the surface area of a sphere of diameter:
(i) 14 cm (ii) 21 cm (iii) 3.5 m

Solution: (i) Here, and

The surface area of a sphere

(ii) Here, and

The surface area of a sphere

(iii) Here, and

The surface area of a sphere

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

Solution:

Solution: Here,

The total surface area of a hemisphere

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.

Solution: Here,

The surface area of the spherical balloon

and

The surface area of the spherical balloon

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 .

Solution: Here, , and

The surface area of a hemisphere bowl

Therefore, the cost of tin-plating on the inside of a hemisphere bowl

6. Find the radius of a sphere whose surface area is 154 cm².

Solution: let be the radius of a sphere .

A/Q,

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

Solution: Given ,

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.

Solution: Here, and

The outer curved surface area of the hemispherical 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).

Solution: Here, the radius of the sphere is .

(i) The surface area of the sphere
(ii) Here,

The curved surface area of the cylinder

(iii) The ratio of the surface area of the sphere and the cylinder :

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?

Solution: Here, ,and

The volume of a cuboidal match box

Therefore, the volume of 12 cuboidal match box

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)

Solution: Here, ,and

The volume of a cuboidal match box

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?

Solution: let be the high of the cuboidal vessel .

Here, ,

The volume of a cuboidal vessel

A/Q,

Therefore, the high of the cuboidal vessel is 4.75 m .

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

Solution: Here, ,and

The volume of a cuboidal pit

Therefore, the cost of digging a cuboidal pit

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.

Solution: let be the high of the cuboidal tank .

Here, ,

The volume of a cuboidal tank

Given , the capacity of a cuboidal tank

A/Q,

Therefore, the high of the cuboidal tank is 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?

Solution: let be the number of day .

Here, ,and

The volume of a tank

A/Q,

Therefore, the number of days is 3 .

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.

Solution: For cuboidal godown :

Here, ,and

The volume of a cuboidal godown

For wooden crates : Here, ,and

The volume of a cuboidal godown

Therefore, the maximum number of wooden crates

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.

Solution: Here,

The volume of the solid cube

The volume of a new cube

Let be the side of a new cube .

A/Q ,

Therefore, the side of a new cube is 6 cm .

Second part : Here ,

The surface area of a cube

Here,

The surface area of a new cube

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?

Solution: Here, ,

and the speed of the water

The length

The volume of the water in the river

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

)

Solution: Here,

A/Q,

The volume of a cylindrical vessel

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.

Solution: Here, , ,

, ,

The volume of a cylindrical wooden pipe

Therefore, the mass of the pipe

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?

Solution: (i) Here ,,

The volume of a rectangular drink pack

(ii) Here, , and

The volume of a plastic cylinder

Therefore, the cylinder has the greater capacity by 85 (= 385 – 300) cm³ .

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)

Solution: (i) Here,

A/Q,

(ii) Here, ,

The volume of a cylinder

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.

Solution: (i) The inner curved surface area of the vessel

(ii) Here,

let be the radius of the base .

A/Q ,

(iii) Here, ,

The volume of a cylindrical 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?

Solution: Here,

A/Q, The volume of a cylindrical vessel = 15.4 litre

Therefore, the surface area of a closed cylindrical vessel

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.

Solution: Here, ,

, ,

The volume of the wood

The volume of a 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?

Solution: Here, , and

The volume of a cylindrical bowl

The volume of 250 cylindrical bowl

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

Solution: (i) Here, ,

The volume of the right circular cone

(ii) Here, ,

The volume of the right circular cone

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 .

Solution: (i) Here, ,

The volume of a conical cone

(ii) Here, ,

The volume of a conical cone

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

, find the radius of the base. (Use π = 3.14)

Solution: let be the radius of the base.

Here,

A/Q,

4. If the volume of a right circular cone of height 9 cm is

, find the diameter of its base.

Solution: let be the radius of the cone .

Here,

A/Q,

Therefore, the diameter of the cone is 8 cm .

5. A conical pit of top diameter 3.5 m is 12 m deep. What is its capacity in kilolitres?

Solution: Here, , ,

The volume of the conical pit

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 .

Solution: (i) Here, ,

A/Q,

(ii) Here, ,

The slant height

(iii) Here, ,

The 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.

Solution: In given figure :

Here , ,

The volume of the solid

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.

Solution: In given figure :

Here , and

The volume of the solid

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.

Solution: Here, , and

The volume of a cone

Again , Here ,

The slant height

The area of the canvas

(Approx.)

EXERCISE 13.8

Assume , unless stated otherwise.

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

Solution: Here,

The volume of a sphere

(ii) Here,

The volume of a sphere

(approx)

2. Find the amount of water displaced by a solid spherical ball of diameter (i) 28 cm (ii) 0.21 m

Solution: (i) Here, ,

The volume of a solid spherical ball

(ii) Here, ,

The volume of a solid spherical ball

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

Solution: Here, ,

The volume of a metallic ball

Therefore, the mass of the ball

(appox.)

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?

Solution: For earth : Here,

The volume of the earth

For earth : Here,

The volume of the earth

Given,

Volume of moon : Volume of earth =1 : 64 .

5. How many litres of milk can a hemispherical bowl of diameter 10.5 cm hold ?

Solution: Here, and

The volume of the hemispherical bowl

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.

Solution: Here, ,

and

The volume of a hemispherical tank

7. Find the volume of a sphere whose surface area is 154

Solution: let be radius of a sphere .

A/Q,

The volume of a sphere

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.

Solution: (i) Total cost = Rs 4989.60

The inside surface area of the dome

(ii) let be the radius of a dome .

A/Q, The surface area of the dome

The 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′.

Solution: (i) Given, and are radius of a small sphere and new sphere respectively .

The volume of a sphere

Again, The volume of a new sphere

A/Q,

(ii) Given ,

The surface area of small sphere

And The surface area of small sphere

A/Q,

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?

Solution: Here, ,

The volume of a capsule (sphere)

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 ?