Surface Areas and Volumes Class 10

Ex 12.1, 1 teackoo
2 cubes each of volume 64 cm? are joined end to end. Find the
surface area of the resulting cuboid.
xs
<
iz
Ceo Be)
—_— — Length=a+a
a a
It is given that
Volume of 1 cube = 64 cm?
(Side)? = 64 cm?
a3 = 64 cm?
a=.
a=4
So,a=4cm

Ex 12.1, 2 teackoo
A vessel is in the form of a hollow hemisphere mounted by a hollow
cylinder. The diameter of the hemisphere is 14 cm and the total heigh
of the vessel is 13 cm. Find the inner surface area of the vessel.
Now, 13cm
Inner surface area of the vessel
= Curved surface area of hemisphere tL
+ Curved surface area of the cylinder
Curved Surface area of hemisphere
Diameter of hemisphere = 14 cm
+ + 14
Radius of hemisphere = r = > 27 em

Ex 12.1, 3 teackoo
A toy is in the form of a cone of radius 3.5 cm mounted ona
hemisphere of same radius. The total height of the toy is 15.5 cm.
Find the total surface area of the toy. ok Soleil
Total surface area of toy 15.5

= Curved Surface area of hemisphere

+ Curved Surface area of cone _4__s
Curved Surface area of hemisphere
Radius of hemisphere = Radius of cone =r = 3.5 cm
Curved surface area of hemisphere = 2777
=2x=x (3.5)?

Ex 12.1, 4 teackoo
A cubical block of side 7 cm is surmounted by a hemisphere. What
is the greatest diameter the hemisphere can have? Find the surface
area of the solid.
The hemisphere can occupy
whole of the side of cube. To
Hence, 7cm

Greatest diameter of hemisphere = Side of cube = 7 cm
Here, base of hemisphere falls on cube, so that Area should not
from part of solid.

Ex 12.1, 5 teackoo
A hemispherical depression is cut out from one face of a cubical
wooden block such that the diameter / of the hemisphere is equal to
edge of the cube. Determine the surface area of the remaining solid.
Given that

Diameter of the hemisphere is

equal to the edge of the cube !
So, Diameter = Side of cube =/

I

Here, base of hemisphere would not be included in the
total solid area of wooden cube.

Ex 12.1, 6 teackoo
A medicine capsule is in the shape of a cylinder with two
hemispheres stuck to each of its ends (see figure}. The length of the
entire capsule is 14 mm and the diameter of the capsule is S mm.
Find its surface area.
Reis Ra
Total area of the capsule
= Curved Surface area of cylinder

+ Curved Surface area of 2 hemispheres
Curved Surface area of cylinder
Diameter of cylinder = 5mm
Radius of cylinder = — = . mm

Ex 12.1,7 teachoo
A tentis in the shape of a cylinder surmounted by a conical top. If
the height and diameter of the cylindrical part are 2.1 mand4m
respectively, and the slant height of the top is 2.8 m, find the area
of the canvas used for making the tent. Also, find the cost of the
canvas of the tent at the rate of Rs 500 per m2. (Note that the base
of the tent will not be covered with canvas.) 2.3m
To find cost, we need area of canvas
Now, a4
Area of the canvas

= Curved area of cone + Curved surface area of the cylinder

Ex 12.1,8 teackoo
From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a
conical cavity of the same height and same diameter is hollowed
out. Find the total surface area of the remaining solid to the
nearest cm?. 14cm
—_—

Since cylinder is solid, it has a base
also whose area is also to be calculated. 2.4 cm
Now,
Total surface area of remaining solid

= Curved Surface Area of cylinder + Area of cylinder base

+ Curved surface area of cone

Ex 12.1, 9 teackoo
A wooden article was made by scooping out a hemisphere from
each end of a solid cylinder, as shown in figure. If the height of the
cylinder is 10 cm, and its base is of radius 3.5 cm, find the total
surface area of the article. r= 3.5 cm
IS
Now,
{surf F 10 cm
Total surface area of article
MU
= Curved surface area of cylinder QUID
+2 x Curved surface area of hemisphere
Curved surface area of cylinder
Radius =r =3.5 cm
Height of the cylinder = h= 10cm

Ex 12.2, 1 teackoo
A solid is in the shape of a cone standing on a hemisphere with
both their radii being equal to 1 cm and the height of the cone is
equal to its radius. Find the volume of the solid in terms of m1.
Now,
Total volume of the solid
= Volume of cone + Volume of hemisphere
Volume of cone
Here, Radius = r=1cm m
& Height = h=1cm
Volume of cone = s7r’h

&x12.2,2 teachoo
Rachel, an engineering student, was asked to make a model shaped
like a cylinder with two cones attached at its two ends by using a thin
aluminium sheet. The diameter of the model is 3 cm and its length is
12 cm. If each cone has a height of 2 cm, find the volume of air
contained in the model that Rachel made. (Assume the outer and
inner dimensions of the model to be nearly the same.)
Since the model contains cylinder and 2 cones. 2c
Volume of air contained in model
= Volume of model 12 ¢
= Volume of cylinder + Volume of 1° cone
+ Volume of 2™ cone
ce

Ex 12.2, 3 teackhoo
Gulab jamun, contains sugar syrup up to about 30% of its volume.
Find approximately how much syrup would be found in 45 gulab
jamuns, each shaped like a cylinder with two hemispherical ends
with length 5 cm and diameter 2.8 cm (see figure)
<a>
Cee
7 . Gee
Lets first find Volume of 1 gulab jamun oe
Since gulabjamun contains one cylinder and 2 hemispheres
Volume of gulab jamun
= Volume of cylinder + Volume of 2 hemispheres

Ex 12.2, 4 teackoo
A pen stand made of wood is in the shape of a cuboid with four
conical depressions to hold pens. The dimensions of the cuboid are
15 cm by 10 cm by 3.5 cm. The radius of each of the depressions is
0.5 cm and the depth is 1.4 cm. Find the volume of wood in the entir
stand (see figure)
Since cones are made inside the cuboid pen stand
Thus, KP

Volume of wood in stand = Volume of cuboid — Volume of 4 cones
Volume of cuboid
Length =/=15cm
Breadth = b=10cm
Height = h = 3.5 cm

Ex 12.2, 5 teackoo
A vessel is in the form of an inverted cone. Its height is 8 cm and
the radius of its top, which is open, is 5 cm. It is filled with water up
to the brim. When lead shots, each of which is a sphere of radius
0.5 cm are dropped into the vessel, one-fourth of the water flows
out. Find the number of lead shots dropped in the vessel.
Radius = 5 cm
Number of lead shots = “2™™< of water flown out
Volume of 1 lead shot

8cm

Volume of water flown out Tee
Volume of water flown out =; x Volume of cone

Ex 12.2, 6 teackoo
A solid iron pole consists of a cylinder of height 220 cm and base
diameter 24 cm, which is surmounted by another cylinder of height
60 cm and radius 8 cm. Find the mass of the pole, given that 1 cm?
of iron has approximately 8g mass. (Use m = 3.14)
r=8cm
>
To find mass of pole, h=60cm
we need to find volume of pole
Ps by
24cm

Now, h = 220 cm
Volume of pole

= Volume of small cylinder

+ Volume of large cylinder

Ex 12.2, 7 teachoo
A solid consisting of a right circular cone of height 120 cm and radius
60 cm standing on a hemisphere of radius 60 cm is placed upright in
a right circular cylinder full of water such that it touches the bottom.
Find the volume of water left in the cylinder, if the radius of the
cylinder is 60 cm and its height is 180 cm.
r=60cm
Volume of water left in cylinder A
= Volume of cylinder — Volume of solid ' a
h=180cm|| 4 | s
Volume of cylinder , : med
Radius =r =60cm L op /
Height = h= 180 cm
Volume of outer cylinder = 2r2h

Ex 12.2, 8 teackoo

A spherical glass vessel has a cylindrical neck 8 cm long, 2 cm in

diameter; the diameter of the spherical part is 8.5 cm. By

measuring the amount of water it holds, a child finds its volume to

be 345 cm?. Check whether she is correct, taking the above as the

inside measurements, and m= 3.14. 2cm

h=8c

Now,

Volume of vessel ;
= Volume of sphere + Volume of cylinder

Volume of sphere

Diameter = 8.5 cm

Radius =r = — = = = 4.25 cm

Example 1 teachoo
Rasheed got a playing top (lattu) as his birthday present, which
surprisingly had no colour on it. He wanted to colour it with his crayon
The top is shaped like a cone surmounted by a hemisphere(see figure}.
The entire top is 5 cm in height and the diameter of the top is 3.5 cm.
Find the area he has to colour. (Take n= =) -4q---
Now, 5cm
Surface area to colour

= Surface Area of hemisphere a.

+ Curved Surface Area of cone

Surface Area of hemisphere
Diameter of hemisphere = 3.5 cm

Example 2 teackoo
The decorative block shown in figure is made of two solids — a
cube and a hemisphere. The base of the block is a cube with edge 5
cm, and the hemisphere fixed on the top has a diameter of 4.2 cm.
Find the total surface area of the block.(Taken = =) a
ct om >
Now, ay
Total surface area of block 5em
= Total Surface area of cube

+ Curved Surface area of hemisphere Sem

— Base area of hemisphere
Total Surface area of cube
Given that side of cube is Scm

Example 3 teachoo
A wooden toy rocket is in the shape of a cone mounted ona cylinder,
as shown in figure. The height of the entire rocket is 26 cm, while the
height of the conical part is 6 cm. The base of the conical portion has
a diameter of 5 cm, while the base diameter of the cylindrical portion
is 3. cm. If the conical portion is to be painted orange and the
cylindrical portion yellow, find the area of the rocket painted with
each of these colours. (Take m= 3.14)
Ne
Now, 26 em
Area to be painted orange
= Curved surface area of the cone @ cm
+ Base area of the cone
base of cylinder
— Base area of the cylinder base of cone

Mayank made a bird-bath for his garden in the shape of a cylinder
with a hemispherical depression at one end (see figure). The height
of the cylinder is 1.45 m and its radius is 30 cm. Find the total
surface area of the bird-bath.(Take n = 2) 30cm
7 a>
EY] 145m

Now,

Total surface area of the bird-bath | | |

= Curved Surface area of cylinder
+ Curved Surface area of hemisphere

Curved Surface area of cylinder
Curved Surface area of cylinder = 27zrh

teackhoo
Example 5
Shanta runs an industry in a shed which is in the shape of a cuboid
surmounted by a half cylinder (see figure ). If the base of the shed is
of dimension 7 m x 15 m, and the height of the cuboidal portion is 8
m, find the volume of air that the shed can hold. Further, suppose
the machinery in the shed occupies a total space of 300 m?, and
there are 20 workers, each of whom occupy about 0.08 m? space on
an average. Then, how much air is in the shed? Take m = =)
A \<
Total volume :
= Volume of cuboid :
4
+5 Xx Volume of cylinder

Example 6 teackoo
A juice seller was serving his customers using glasses as shown in
figure. The inner diameter of the cylindrical glass was 5 cm, but the
bottom of the glass had a hemispherical raised portion which
reduced the capacity of the glass. If the height of a glass was 10 cm,
find the apparent capacity of the glass and its actual capacity. (Use
m= 3.14.)
Scm

Now,
Apparent capacity of the glass = Volume of cylinder ne
And, Pes
Actual capacity of the glass

= Volume of cylinder — Volume of hemisphere

Example 7 teachoo
A solid toy is in the form of a hemisphere surmounted by a right
circular cone. The height of the cone is 2 cm and the diameter of
the base is 4 cm. Determine the volume of the toy. If a right circular
cylinder circumscribes the toy, find the difference of the volumes of
the cylinder and the toy.(Take m = 3.14)
Now,
Volume of toy = Volume of cone + Volume of hemisphere
A
Volume of cone VAN i cm
|
Height of cone = OA=h=2 cm NO
Diameter of cone = BC = 4cm Le EE
So, Radius = re eg 2cm

For calculations, Lateral Surface Area means curved surface area. It's called lateral surface area for cube and cuboid and curved surface area for cylinder, cone and hemisphere.
The formulas for calculating Surface Area and Volume are
Cube
Lateral Surface Area = 4a
2
Total Surface Area = 6a
2
Volume = a
3
Cuboid
Lateral Surface Area = 2(lh + bh) = 2 (l + b) h
Total Surface Area = 2(lb + bh + lh)
Volume = l x b x h
Cylinder
Curved Surface Area = 2πrh
Total Surface Area = 2πrh + πr
2
+ πr
2
= 2πrh + 2πr
2
= 2πr(r + h)
Volume = πr
2
h
Cone
Curved Surface Area = πrl
Total Surface Area = πrl + πr
2
= πr(r + l)
Volume = 1/3 πr
2
h
Hemisphere
Curved Surface Area = 2πr
2
Total Surface Area = 2πr
2
+ πr
2
= 3πr
2
Volume = 2/3 πr
3
Sphere
Surface Area = 4πr
2
Volume = 4/3 πr
3
Frustum
Curved Surface Area = π(r
1
+ r
2
)l
Total Surface Area =π(r
1
+ r
2
)l + πr
1
2
+ πr
2
2
Volume = 1/3 πh (r
1
2
+ r
2
2
+ r
1
r
2
)

Example 8 teachoo.com
A cone of height 24 cm and radius of base 6 cm is made up of
modelling clay. A child reshapes it in the form of a sphere. Find the
radius of the sphere.

r=6cm
Given that, child reshapes cone with sphere.
So, Volume of sphere = Volume of cone
Volume of cone
Height of cone = h = 24cm
Radius = r=6cm

Example 9 teachoo.com
Selvi’s house has an overhead tank in the shape of a cylinder. This is
filled by pumping water from a sump (an underground tank) which
is in the shape of a cuboid. The sump has dimensions 1.57 m x 1.44
m x 95cm. The overhead tank has its radius 60 cm and height 95
cm. Find the height of the water left in the sump after the overhead
tank has been completely filled with water from the sump which
had been full. Compare the capacity of the tank with that of the
sump. (Use n= 3.14)
= 60cm
h=95 cm
h=95
y- 1.44m
7= 157m
Lets first find volume of water left in sump
Volume of water left in the cuboidal sump after filling the tank
= Volume of cuboidal sump — Volume of cylindrical tank

teachoo.com
Example 10
A copper rod of diameter 1 cm and length 8 cm is drawn into a wire
of length 18 m of uniform thickness. Find the thickness of the wire.
icm r
[.- => h=18m
Volume of copper rod = Volume of wire
Volume of copper rod
Copper rod is in form of cylinder with
Diameter of rod = 1 cm
: 1
So, radius = r = zem
And, length = height = h = 8 cm

Example 11 teachoo.com
A hemispherical tank full of water is emptied by a pipe at the rate
of 3 litres per second. How much time will it take to empty half the
tank, if it is 3m in diameter?(Take n = =)
Volume of tank
Tank is in form of hemisphere with
Diameter = 3m
: 3
So, radius = r = 3m
Volume of the tank = Snr
3
a2y2By G)
3° 7 2
= 2 22 3X3 x3
3° 7 “2x2x2
=» mB
14

Example 12 (Method 1 By deriving frustum formula) teackoo.com
The radii of the ends of a frustum of a cone 45 cm high are 28 cm and
7 cm (see figure). Find its volume, the curved surface area and the
total surface area(Take n = =) , Zan
There are two cones OCD & OAB
4
hy

SAP ff

Volume of frustum ABDC | \ Ok
oO
= Volume of cone OAB — Volume of cone OCD
Curved Surface area of frustum ABDC
= Curved Surface area of cone OAB — Curved Surface area of cone OCD
Total Surface area of frustum ABDC
= Curved Surface area of frustum
+ Area of top circle + Area of bottom circle

Example 13 teachoo.com
Hanumappa and his wife Gangamma are busy making jaggery out
of sugarcane juice. They have processed the sugarcane juice to
make the molasses, which is poured into moulds in the shape of a
frustum of a cone having the diameters of its two circular faces as
30 cm and 35 cm and the vertical height of the mould is 14 cm (see
figure}. If each cm? of molasses has mass about 1.2 g, find the mass
of the molasses that can be poured into each mould. (Take n = =)
To find mass of molasses, a;
we need to find volume of molasses h
30cm
a
Since the mould is in the shape of a frustum of a cone.
Volume of molasses = Volume of frustum
1
= gmn(r,? +122 + F419)

Example 14 teachoo.com
An open metal bucket is in the shape of a frustum of a cone,
mounted on a hollow cylindrical base made of the same metallic
sheet (see Fig. 13.23). The diameters of the two circular ends of the
bucket are 45 cm and 25 cm, the total vertical height of the bucket
is 40 cm and that of the cylindrical base is 6 cm. Find the area of the
metallic sheet used to make the bucket, where we do not take into
account the handle of the bucket. Also, find the volume of water
the bucket can hold.{Take nm = =)
, 7
Area of metallic sheet used a
= curved surface area of frustum
. 40cm
+ curved surface area of cylinder
25cm
+ area of circular base Ts 6c

Adventure camps are the perfect place for the children to practice decision making for themselves without parents and teachers guiding their every move. Some students of a school reached for adventure at Sakleshpur. At the camp, the waiters served some students with a welcome drink in a cylindrical glass and some students in a hemispherical cup whose dimensions are shown below. After that they went for a jungle trek. The jungle trek was enjoyable but tiring. As dusk fell, it was time to take shelter. Each group of four students was given a canvas of area 551m2. Each group had to make a conical tent to accommodate all the four students. Assuming that all the stitching and wasting incurred while cutting, would amount to 1m2, the students put the tents. The radius of the tent is 7m.
Refer to Design A
Question 1
The volume of cylindrical cup is
(a) 295.75 cm
3
(b) 7415.5 cm
3
(c) 384.88 cm
3
(d) 404.25 cm
3
Question 2
The volume of hemispherical cup is
(a) 179.67 cm
3
(b) 89.83 cm
3
(c) 172.25 cm
3
(d) 210.60 cm
3
Refer Design B:
Question 3
Which container had more juice and by how much?
(a) Hemispherical cup, 195 cm
3
(b) Cylindrical glass, 207 cm
3
(c) Hemispherical cup, 280.85 cm
3
(d) Cylindrical glass, 314.42 cm
3
Question 4
The height of the conical tent prepared to accommodate four students is
(a) 18 m
(b) 10 m
(c) 24 m
(d) 14 m
Question 5
How much space on the ground is occupied by each student in the conical tent
(a) 54 m
2
(b) 38.5 m
2
(c) 86 m
2
(d) 24 m
2

The Great Stupa at Sanchi is one of the oldest stone structures in India, and an important monument of Indian Architecture. It was originally commissioned by the emperor Ashoka in the 3rd century BCE. Its nucleus was a simple hemispherical brick structure built over the relics of the Buddha. It is a perfect example of combination of solid figures. A big hemispherical dome with a cuboidal structure
mounted on it. (Take π=22/7)
Question 1
Calculate the volume of the hemispherical dome if the height of the dome is 21 m –
(a) 19404 cu. m
(b) 2000 cu. m
(c) 15000 cu. m
(d) 19000 cu. m
Question 2
The formula to find the Volume of Sphere is
(a) 2/3 πr
3
(b) 4/3 πr
3
(c) 4 πr
2
(d) 2 πr
2
Question 3
The cloth require to cover the hemispherical dome if the radius of its base is 14m is ?
(a) 1222 sq.m
(b) 1232 sq.m
(c) 1200 sq.m
(d) 1400 sq.m
Question 4
The total surface area of the combined figure i.e. hemispherical dome with radius 14m and cuboidal shaped top with dimensions 8m × 6m × 4m is
(a) 1200 sq.m
(b) 1232 sq.m
(c) 1392 sq.m
(d) 1932 sq.m
Question 5
The volume of the cuboidal shaped top is with dimensions mentioned in question 4
(a) 182.45 m
3
(b) 282.45 m
3
(c) 292 m
3
(d) 192 m
3

On a Sunday, your Parents took you to a fair. You could see lot of toys displayed, and you wanted them to buy a RUBIK’s cube and strawberry ice-cream for you. Observe the figures and answer the questions-:
Question 1
The length of the diagonal if each edge measures 6cm is
(a) 3√3
(b) 3√6
(c) √12
(d) 6√3
Question 2
Volume of the solid figure if the length of the edge is 7cm is-
(a) 256 cm
3
(b) 196 cm
3
(c) 343 cm
3
(d) 434 cm
3
Question 3
What is the curved surface area of hemisphere (ice cream) if the base radius is 7cm?
(a) 309 cm
2
(b) 308 cm
2
(c) 803 cm
2
(d) 903 cm
2
Question 4
Slant height of a cone if the radius is 7cm and the height is 24 cm___
(a) 26 cm
(b) 25 cm
(c) 52 cm
(d) 62 cm
Question 5
The total surface area of cone with hemispherical ice cream is
(a) 858 cm
2
(b) 885 cm
2
(c) 588 cm
2
(d) 855 cm
2

Ex 13.3, 1 teachoo.com
A metallic sphere of radius 4.2 cm is melted and recast into the
shape of a cylinder of radius 6 cm. Find the height of the cylinder.
r=6cm
Since sphere is melted into a cylinder ,
So, Volume of sphere = volume of cylinder
Volume of sphere
Radius =r=4.2 cm
4
volume of sphere = gar
= m0(4.2)°

Ex 13.3, 2 teachoo.com
Metallic spheres of radii 6 cm, 8 cm and 10 cm, respectively, are melte
to form a single solid sphere. Find the radius of the resulting sphere.
Since 3 spheres are melted to from one new sphere.
Volume of 3 old sphere = volume of new sphere
Volume of 1% sphere + volume of 2"? sphere + volume of 3 sphere

= Volume of new sphere

teachoo.com
Ex 13.3, 3
A 20 m deep well with diameter 7 m is dug and the earth from
digging is evenly spread out to form a platform 22 m by 14 m. Find
the height of the platform.
7m
h=20m — h
ys - 14m
7= 22m
Here, shape of well is in form of cylinder
& shape of platform will be in shape of cuboid.
Hence, Volume of well = Volume of platform
So , Volume of cylinder = Volume of cuboid

Ex 13.3, 4 teachoo.com
A well of diameter 3 m is dug 14 m deep. The earth taken out of it has
been spread evenly all around it in the shape of a circular ring of width
4m to form an embankment. Find the height of the embankment.
3m
OO
Ke
= h
Cylinder A | h=14m =
Both well and embankment are in the from of cylinder.
Let well be cylinder A and embankment be cylinder B.
Since mud of well is distributed in embankment
Volume of well = Volume of embankment

Ex 13.3, 5 teachoocom
A container shaped like a right circular cylinder having diameter 12
cm and height 15 cm is full of ice cream. The ice cream is to be
filled into cones of height 12 cm and diameter 6 cm, having a
hemispherical shape on the top. Find the number of such cones
which can be filled with ice cream.
12 cm

a | - ~ 7

Number of cones = Folume opine cream cone

Ex 13.3, 6 teachoo.com
How many silver coins, 1.75 cm in diameter and of thickness 2 mm,
must be melted to form a cuboid of dimensions 5.5 cm x 10 cm x 3.5
cm?
—>= th = 2mm =>
h=3.5cm
a 4 10 cm
7= 5.5cm
. _ Volume of cuboid
Number of coins = Volume of ico
Volume of cuboid
Length ()=5.5 cm
Breadth (b) = 10 cm
Height (h) = 3.5 cm

teachoo.com
Ex 13.3, 7
A cylindrical bucket, 32 cm high and with radius of base 18 cm, is
filled with sand. This bucket is emptied on the ground and a conical
heap of sand is formed. If the height of the conical heap is 24 cm,
find the radius and slant height of the heap.
r=18cm
h=32cn
4cm
-
Since sand in cylindrical bucket is emptied to make a conical heap
Volume of cylindrical bucket = volume of conical heap

teachoo.com
Ex 13.3, 8
Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed
of 10 km/h. How much area will it irrigate in 30 minutes, if 8 cm of
standing water is needed?
Canal is the shape of cuboid where
ei 10.krn/h

Breadth =6m

Height = 1.5m _
& Speed of canal = 10 km/hr beem
Length of canal in 1 hour = 10 km
Length of canal in 60 minutes = 10 km
Length of canal in 1 min = a x 10km
Length of canal in 30 min = = x 10 = 5km = 5000 m

Ex 13.3, 9 teachoo.com
A farmer connects a pipe of internal diameter 20 cm from a canal
into a cylindrical tank in her field, which is 10 m in diameter and 2
m deep. If water flows through the pipe at the rate of 3 km/h, in
how much time will the tank be filled? A 20 em
| _

Let length of pipe for filling whole tank be h m.
So,

Volume of pipe = Volume of tank

teachoo.com
Ex 13.4,1
A drinking glass is in the shape of a frustum of a cone of height 14
cm. The diameters of its two circular ends are 4 cm and 2 cm. Find
the capacity of the glass.
Since glass is in from of frustum
+ m

Capacity of glass = Volume of frustum

= smh(r? +7? +7472) =

cm
Hence,
h = height of frustum = 14 cm
the Diameter of 15' end a4 em
2 2

the Piameter of 2nd end - 2 =1em

Ex 13.4, 2 teachoo.com
The slant height of a frustum of a cone is 4 cm and the perimeters
(circumference) of its circular ends are 18 cm and 6 cm. Find the
curved surface area of the frustum.
We know that
Curved surface area of frustum = #(r, + r2)l
7=4cm

Here /=4cm
We need to find r, & r,
For r, For r,
Circumference of 1%t end = 18 cm | Circumference of 2"¢ end = 6 cm

2nr,=18 2nr, =6

ne 2 a ne 2 iu

9 3
n= z r= z

Ex 13.4, 3 teachoo.com
A fez, the cap used by the Turks, is shaped like the frustum of a
cone (see figure). If its radius on the open side is 10 cm, radius at
the upper base is 4 cm and its slant height is 15 cm, find the area of
material used for making it
Since the fez (cap) is in from of frustum ne 4 cm
and is covered from upper side. ro 8 a .
7=15cem/f o /\
K ° ay
Area of material used to make cap fs ° ° ba
VaArrB
= Curved surface area of frustum ANS c
+ Area of circle portion of upper side
=a(r, + r.)b+ nr)?
Here
r, = Radius of lower base = 10 cm
r, = Radius of upper base = 4 cm
7 = Slant height = 15 cm

teachoo.com
Ex 13.4, 4
A container, opened from the top and made up of a metal sheet, is
in the form of a frustum of a cone of height 16 cm with radii of its
lower and upper ends as 8 cm and 20 cm, respectively. Find the
cost of the milk which can completely fill the container, at the rate
of Rs 20 per litre. Also find the cost of metal sheet used to make the
container, if it costs Rs 8 per 100 cm*. (Take n= 3.14)
In order to find
Cost of milk which can completely fill container ,
we need to find volume (in litres) Cc 3
Volume of container = Volume of frustum m

= smh(r? + retry.)
an

Here r=8cm
h = height = 16 cm
r, = radius of upper end = 20cm
r, = radius of lower end =8cm

Ex 13.4, 5 teachoo.com
A metallic right circular cone 20 cm high and whose vertical angle is
60° is cut into two parts at the middle of its height by a plane
parallel to its base. If the frustum so obtained be drawn into a wire
of diameter = cm ,find the length of the wire. fe)
- ey ‘.
‘TN
/ do
Let OCD be the metallic cone f 5088
; \
and ABDC be the required frustum 20 mag B
jt \
Since frustum is drawn into wire _——
Volume of frustum ABDC = Volume of cylindrical wire
Volume of frustum ABDC
Volume of frustum = ere +72 + rr)

Ex 13.5, 1 teachoo.com

A copper wire, 3 mm in diameter, is wound about a cylinder whose

length is 12 cm, and diameter 10 cm, so as to cover the curved

surface of the cylinder. Find the length and mass of the wire,

assuming the density of copper to be 8.88 g per cm?,
—__

Given, |

Diameter of the wire = 3 mm 3mm fe 12 cm

10 cm
-. One round of the wire covers aie cm of the height of the cylinder.
Also, given,
Length of cylinder = 12cm

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Ex 13.5, 2
A right triangle, whose sides are 3 cm and 4 cm (other than
hypotenuse) is made to revolve about its hypotenuse. Find the
volume and surface area of the double cone so formed. (Choose
value of mas found appropriate.) A
Let ABC be the right triangle

where AB = 3cm 3om

and BC=4cm Bo oacm c
Since AC is the hypotenuse

AC? = AB? + BC?

AC? = 37 + 4?

AC?=9 +16

AC? = 25

Ex 13.5, 3 teackoo.com
A cistern, internally measuring 150 cm x 120 cm x 110 cm, has
129600 cm? of water in it. Porous bricks are placed in the water until
the cistern is full to the brim. Each brick absorbs one-seventeenth of
its own volume of water. How many bricks can be put in without
overflowing the water, each brick being 22.5 cm x 7.5cmx 6.5 cm?
Given,
Volume of cistern = 150cm x 120cm x 110cm
= 1980000 cm?
Volume of water = 1296000 cm?
Volume of 1 brick = 22.5 cmx7.5cmx6.5cm
= 1096.875 cm?

Ex 13.5, 4 (Introduction) teackoo.com
In one fortnight of a given month, there was a rainfall of 10 cmina
river valley. If the area of the valley is 97280 km2, show that the total
rainfall was approximately equivalent to the addition to the normal
water of three rivers each 1072 km long, 75 m wide and 3 m deep.
Valley before rainfall Valley after rainfall of 10 cm
(A 10 em
4
4
¢
+ 4
7
7
L/S
Now,
Area of the Valley Volume of rainfall
= Volume of cuboid
= Areax 10cm?

Ex 13. 5 5 teackoo.com
An oil funnel made of tin sheet consists of a 10 cm long cylindrical
portion attached to a frustum of a cone. If the total height is 22 cm,
diameter of the cylindrical portion is 8 cm and the diameter of the
top of the funnel is 18 cm, find the area of the tin sheet required to
make the funnel (see Fig. 13.25)
18 cm
<>
Area of tin sheet
= CSA of Cylinder + CSA of frustum of cone 22cm
10 cm
—>
88cm
Cylinder Frustum of cone
. . 8
Radius of cylinder = ia 4cm Height of frustum = 22cm - 10cm
Height of cylinder = 10cm =l2cm

Ex 13.5, 6 feachoo.com
Derive the formula for the curved surface area and total surface area
of the frustum of a cone, given to you in Section 13.5, using the
symbols as explained.
There are two cones OCD & OAB , ————,
We are given |
i
Height of frustum =h 4
h
Slant height of frustum = / ' Pane
AT DA Ff
Radius PB =r, ‘ef Aa
Radius QD =r, (0)
We need to find
Curved Surface Area & Total Surface Area
Here,
We need to write h,, /,, hy, /, in terms of h and/

teachoo.
Ex 13.5, 7 PAmO One
Derive the formula for the volume of the frustum of a cone, given to
you in Section 13.5, using the symbols as explained.
There are two cones OCD & OAB , “nT a
We are given
Height of frustum = h u) '
h
Slant height of frustum = / ' atk
. AT DA Ff
Radius PB =r, “Q 7, hy
Radius QD =r, cree MR £2
We need to find
Curved Surface Area & Total Surface Area
Here,
We need to write h,, /,, hy, /, in terms of h and /