13. SURFACE AREAS AND VOLUMES (NCERT)
NCERT
13. SURFACE AREAS AND VOLUMES
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Chapter 13. Surface area and volumes |
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Exercise 13.1 complete solution Exercise 13.2 complete solution Exercise 13.3 complete solution Exercise 13.4 complete solution Exercise 13.5 complete solution |
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1. Surface area of Cube , Cuboid , Cylinder , Cone , Sphere and Hemisphere : (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 of Cube , Cuboid , Cylinder , Cone , Sphere and Hemisphere : (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 7. The formulae involving the frustum of a cone are : |
.
.
.
where 
.
,where 
vertical height of the frustum, 
slant height of the frustum , 
and 
are radii of the two bases (ends) of the frustum.EXERCISE 13.1
Unless stated otherwise, taken 
1. 2 cubes each of volumes 64 
are joined end to end . Find the surface area of the resulting cuboid .
Solution: let 
be the length of the cube .
A/Q, 
For cuboid : Here , 
The surface area of cuboid 
2. 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 height of the vessel is 13 cm . Find the inner surface area of the vessel .
Solution: Here, Radius 
The height of the cylinder 
Area of the inner surface of the vessel
Area of cylinder + Area of hemisphere
3. A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius . The total height of the toy is 15.5 cm . Find the total surface area of the toy .
Solution: For cone : Radius 
3.5 cm 
cm ,
Height 
cm
The slant height 
12.5 cm
The curve surface area of cone 
For hemisphere : Radius 
The curve surface area of hemisphere 
The total surface area of the toy 
4. 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 .
Solution: The cubical block of side is 7 cm .
So, the greatest diameter of the hemisphere is 7 cm .
Here , Radius 
, 
The surface area of the solid
Area of cubical block + Area of hemisphere – Area of circular top
5. A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter 
of the hemisphere is equal to the edge of cube . Determine the surface area of the remaining solid .
Solution: Here, diameter 
and Radius
The surface area of the remaining solid
Area of cubical block + Area of hemisphere – Area of circular top
6. A medicine capsule is in the shape of a cylinder with two hemisphere stuck to each of its ends (see Fig 13.10) . The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm .Find its surface area .
Solution: Here, Diameter 
Radiusr 
And height of cylinder 
The surface area of the capsule
S.A. of cylinder + Area of 2 hemisphere
7. A tent is in the shape of a cylinder surmounted by a conical top . If the height and diameter of the cylindrical part are 2.1 m and 4 m 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 
. (Note that the base of the tent will not be covered with canvas)
Solution: Here , Height 
, Diameter 
, Radius 
and the slant height 
The area of the canvas of the tent
Area of the cylinder + C.S. area of cone
The cost of the canvas of the tent 
.
8. 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 
.
Solution: Here , Diameter
,
Radius
, Height
The slant height 
The total surface area of the remaining solid
C.S.A of cylinder + C.S.A. of cone + Area of circular top
[appro.]
9. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig. 13.11 . 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 .
Solution: Here , Radius 
Height
The total surface area of the article
C.S.A. of cylinder + Area of 2 hemisphere
EXERCISE 13.2
Unless stated otherwise , take
1. 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 .
Solution: Given , 
The volume of solid The volume of cone + The volume of hemisphere
2. 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.)
Solution:
3. A 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 liked a cylinder with two hemispherical ends with length 5 cm and diameter 2.8 cm (see Fig. 13.15)
Solution:
4. 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 15cm by 10cm 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 entire stand (see Fig. 13.16) .
Solution: Here , length 
cm , breadth 
cm , height 
cm
The surface area of a cuboidal solid
cm²
cm²
cm²
Therefore, the surface area of cylindrical block
Thus, the surface area of the remaining block
5. 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 .
Solution: For cone : Here,
Height of cone 
cm and Radius 
cm
The volume of cone 
For Sphere : Here, Radius 
cm
The volume of the sphere 
The volume of the water that flows out of the cone
(the volume of the cone)
Therefore, the number of lead shots 
6. 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 
of iron has approximately 8g mass . (Use 
)
Solution : For big cylinder : Here ,
Diameter 
, Radius 
and Height 
The volume of the big cylindrical pole 
For small cylinder : Here ,
Radius 
, Height 
The volume of the big cylindrical pole
The volume of solid pole 
The mass of the pole 
7. 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 .
Solution: For cone and hemisphere : Here,
Radius 
and Height 
The volume of solid
The volume of cone + The volume of hemisphere
For Cylinder :
Here , Radius and Height
The volume of Cylinder
The volume of water left in the cylinder
Volume of Cylinder – Volume of solid
8. 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 find its volume to be 345 . Check whether she is correct , taking the above as the inside measurements, and .
Solution: For cylindrical neck :
Here , 
The volume of cylindrical neck
For spherical part :
Here , 
The volume of spherical part 
The volume of spherical glass vessel
She is not correct . The correct answer is 
.
EXERCISE 13.3
Take , unless stated otherwise.
1. 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 .
Solution:
2. Metallic spheres of radii 6 cm , 8 cm and 10 cm , respectively, are melted to form a single solid sphere . Find the radius of the resulting sphere .
Solution: Here , 
; 
; 
and let 
be the new radius of the sphere .
A/Q , 
cm
3. A 20m deep well with diameter 7 m is dug and the earth from digging is evenly spread out to form a platform 22m by 14m . Find the height of the platform .
Solution:
4. A well of diameter 3 m is dug 14 m deep . The earth taken out it has been spread evenly all around it in the shape of a circular ring of width 4 m to form an embankment . Find the height of the embankment .
Solution:
5. 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 .
Solution: For cylinder : diameter 
cm , Radius 
cm and Height 
cm
The volume of cylinder 
cm³
For cone : Height 
cm , Diameter 
cm and Radius 
cm
The total volume 
The volume of cone 
The volume of hemisphere 
cm³
The total number of such cones which can be filled with ice cream 
10
6. 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 × 10 cm × 3.5 cm ?
Solution:
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 cinical heap is 24 cm , find the radius and slant height of the heap .
Solution: 38. For cylindrical bucket : Here, 
cm and 
cm
The volume of cylindrical bucket 
cm³
For conical heap : Here, 
cm
The volume of conical heap 
cm³
A/Q , The volume of cylindrical bucket The volume of conical heap
cm
The slant height of the heap 
cm
8. Water in a canal , 6m 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 ?
Solution: Since the speed of the water 
km/h 
m/h 
m/h .
So, the length of canal × 
m
Here, 
m , 
m and 
m
The volume of the canal (Cuboid structure) 
m³ .
let 
m² be area is irrigate in 30 minutes .
A/Q , × 
m² 
56.25 hectare . [ 1 hectare = 10000 m2 ]
9. 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 2m deep . If water flows through the pipe at the rate of 3 km/h , in how much time will the tank be filled ?
Solution: let it filled the tank in 
hours .
For pipe : Here , Diameter 
cm and Radius 
cm cm 
m
The length of pipe 
km/hr × = 4000 m × 
The volume of pipe 
× 
× 
cm³
For cylindrical tank : Here , Diameter 
m , Radius 
m 5m and 
m
The volume of cylindrical tank × 5 × 5 × 2 
cm³
A/Q , The volume of pipe = The volume of cylindrical tank
× × 
× 
hrs × 60 5 × 15 75 minutes
EXERCISE 13.4
Use unless stated otherwise.
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 .
2. 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.
3. A fez , the cap used by the Turks , is shaped like the frustum of a cone (see Fig. 13.24) . 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 .
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 
. (Take )
Solution: Here, 
cm , 
cm and 
cm
The volume of frustum of a cone 
cm³
cm³
cm³
cm³
= 
litre
The cost of the milk at the rate of Rs. 20 per litre 
The slant height 
cm
Again, the total surface area of a frustum of a cone 
cm²
cm²
The cost of metal sheet used to make the container 
× 
5. 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 .
Solution: Here, the height 
cm
cm , 
, 
, 
and 
In 
We have, 
cm
And 
We have, 
cm
Let, 
be length of the wire .
The diameter of wire 
cm , Radius 
cm.
A/Q , the volume of a cylindrical wire the volume of a frustum of a cone.
[ 1m = 100 cm]
Therefore, the length of the wire is 
m
EXERCISE 13.5 (Optional)*
1. 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 
.
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 π as found appropriate.)
3. A cistern, internally measuring 150 cm × 120 cm × 110 cm, has 129600 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 × 7.5 cm × 6.5 cm?
4. In one fortnight of a given month, there was a rainfall of 10 cm in a river valley. If the area of the valley is 7280 
, 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.
5. 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).
6. 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.
7. Derive the formula for the volume of the frustum of a cone, given to you in Section 13.5, using the symbols as explained.
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