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VOLUMES & SURFACE AREA
FORMULAE:
1. Cuboid ⎯ Let, length = l, breadth = b and height = h units.
(i) Volume of Cuboid = (l×b×h) cubic units.
⎯ Let each edge (or side) of a cube be a units, then ⎯
2. Cube
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(i) Volume of Cube = a3 cubic units.
(ii) Whole surface of cube = 6a2 sq.units.
( )
(iii) Diagonal of cuboid = a 3 units.
Let the radius of the base of a cylinder be r units and its height (or length) be
h units, then
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3. Cylinder ⎯
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l2 + b 2 + h 2 units.
(iii) Diagonal of cuboid =
m
(ii) Whole surface of cuboid = 2(lb + bh + lh) sq.units.
(i) Volume of the cylinder = (πr2h) cubic units.
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(ii) Curved surface area of the cylinder = (2πrh) sq. units
(iii) Total surface area of the cylinder = (2πrh + 2πr2h) sq.units.
4. Sphere
⎯ Let the radius of a sphere be r units, then ⎯
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⎛ 4 3⎞
(i) Volume of the sphere = ⎜ πr ⎟
⎝3
⎠
.s
(ii) Surafce area of the sphere = (4πr2) sq. units.
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⎛ 2 3⎞
(iii) Volume of a hemisphere = ⎜ πr ⎟ cubic units.
⎝3
⎠
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(iv) Curved surface area of the hemisphere = (2πr2) square units.
(v) Whole surface area of the hemisphere = (2πr2 + πr2)
= (3πr2) square units.
5. Cone
⎯ Let r be the radius of the base, h the height and l the slant height of a cone, then–
(i) Slant height l =
h2 + r 2
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⎛1 2 ⎞
(ii) Volume of the cone = ⎜ πr h ⎟ cu. units.
⎝3
⎠
(
) sq. units
m
(iii) Curved surface area of the cone = (πrl) sq. units = πr r 2 + h 2
The surface area of a cube is 384 cm2. Its volume is?
1.
1) 525 cm3
2) 625 cm3
3) 512 cm3
Ans: 3
1) 521 cm3
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Sol: 6a2 = 384 ⇒ a2 = 64
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PROBLEMS
⇒a=8
... Volume = (8×8×8) cm3
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= 512 cm3
The diagnal of a cube is 4 3 m. Its volume is?
2.
2) 89 m2
Ans: 3
3) 100 m2
4) 96 m2
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1) 86 m2
Sol: Let the edge of the cube be a metres
Then
3a
=
4 3 or a = 4
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... Edge = 4 m
.s
So, volume = (4×4×4)m2 =96 m2
3.
What is the length of the longest pole that can be placed in a room 5 m long, 4 m broad
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and 3 m high.
2) 7.50 m
3) 7.00 m
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1) 7.05 m
Ans: 1
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Sol: Length of longest pole = Length of diagonal.
=
52 + 42 + 32 m
=
50 m = 5 2 m
= (5×1.41) m = 7.05 m
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4) 7.07 m
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4.
What is the volume, surface area and the length of diagonal of a cuboid 12 m long, 9
m broad and 8 m high?
1) 15 m
2) 17 m
3) 16 m
4) 18 m
Ans: 2
m
Sol: Volume = (12×9×8)m3= 864 m3
(12 ) + (9 ) + (8 )
2
Diagonal =
2
289 m = 17 m.
=
Three cubes of sides 6 cm, 8 cm and 1 cm are melted to form a new cube. What is the length of
the edge of the new cube.
1) 27 cm
2) 18 cm
3) 36 cm
Ans: 4
= 729cm3
1
= 9 cm
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... Edge of new cube = (9 × 9 × 9 )3
4) 9 cm
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3
3
3
Sol: Volume of new cube = ⎡⎣(6 ) + (8 ) + (1) ⎤⎦ cm3
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5.
2
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2
2
Surface area = 2× (12×9 + 9×8 + 12×8) m = 552 m
The capacity of a cylindrical tank is 6160 m3. If the radius of its base is 14 m, what is
6.
1) 9 m
3) 12 m
4)16 m
22
× 14×h = 6160
7
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Sol:
2) 10 m
.s
Ans: 2
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the depth of the tank?
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7
1 ⎞
⎛
... h = ⎜ 6160 × ×
⎟ = 10 m
22 14 × 14 ⎠
⎝
7.
The volume of a sphere is 38808 cu. cm. The curved surface area of the sphere is?
1) 5544 sq. cm
2) 1386 sq. cm
3) 8316 sq. cm
Ans: 1
⎛ 4 3⎞
Sol: Given the volume of the sphere: ⎜ πr ⎟
⎝3
⎠
= 38808
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4) 4158 sq. cm
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7 3⎞
⎛
⇒ r3 = ⎜ 38808 × ×
22 4 ⎠
⎝
= (21)3
r = 21 cm
So, the curved area of sphere = 4πr
m
22
⎛
⎞
2
⎜ 4 × × 21× 21⎟ cm
7
⎝
⎠
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=
2
8.
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= 5544 cm2
The length of cylinder is 80 cm and the diameter of its base is 7 cm. The whole surface
of the cylinder is?
1) 1837 cm2
2) 1760 cm2
3) 3080 cm2
Ans: 1
du
c
Sol: Given diameter = 7 cm, so radius r = 3.5 cm
4) 1942 cm2
And h = 80 cm.
= (2πrh + 2πr2h) = 2πr(h + r)
∴ Whole surface
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= ⎛⎜ 2 × 22 × 3.5 × (80 + 3.5) ⎞⎟ cm2
7
⎝
⎠
= 22×83.5
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= 1837 cm2
9.
If the volumes of two cones are in the ratio 1:4 and their diameters are in the ratio 4:5,
.s
then the ratio of their heights is?
2) 5:4
3) 5:16
4) 25:64
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1) 1:5
Ans: 4
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Sol: Since the diameters are in the ratio 4:5. It follows that their radii are in the ratio 4:5.
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Let them be 4r and 5r. Let the heights be h and H
Ratio of volumes
1
2
π × (4r ) × h
3
= 1
2
π × (5r ) × H
3
=
16 h
16 h
⇒
25 H
25 H
h : H = 25: 64
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=
1
4
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10. If a right circular cone of vertical height 24 cm has a volume of 1232 cm3, then the
area of its curved surface is?
1) 1254 cm2
Ans: 3
2) 704 cm2
3) 550 cm2
4) 154 cm2
m
Sol: Given volume of the cone is 1232 cm3
Now,
r = 7 cm.
and
h = 24
=
7 2 + (24) 2
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l=
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7
3 ⎞
⎛
r2 = ⎜ 1232 × × ⎟ = 49
22 24 ⎠
⎝
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⎛ 1 22 2
⎞
⎜ × × r × 24 ⎟ = 1232
⎝3 7
⎠
625 = 25 cm
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∴ Curved surface area = πrl
2
⎛ 22
⎞
= ⎜ × 7 × 25 ⎟ cm2 = 550 cm
⎝ 7
⎠
EXERCISE
A wooden box of dimensions 8 m × 7 m × 6 m is to carry rectangular boxes of
ak
1.
dimensions 8 cm × 7 cm × 6 cm. The maximum number of boxes that can be carried
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in 1 wooden box is?
2) 1000000
3) 9800000
4) 7500000
The sum of length, breadth and depth of a cuboid is 19 cm and its diagonal is 5 5
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2.
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1) 1200000
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cm. Its surface area is ?
3.
1) 361 cm2
2) 125 cm2
3) 236 cm2
4) 486 cm2
If the length of diagonal of a cube is 8 3 cm, then its surface area is?
1) 512 cm2
2) 384 cm2
3) 192 cm2
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4) 768 cm2
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4.
The length of longest rod that can be placed in a room 20 m long, 16 m broad and 12
m high, is?
1) 20 m
3) 48 m
4) 28.2 m
2) Becomes 4 times
3) Becomes 6 times
4) Becomes 8 times
The curved surface area of a sphere is 1386 sq. cm. Its volume is ?
1) 2772 cm3
7.
2) 4158 cm3
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6.
1) Is doubled
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If each side of a cube is doubled, then its volume is ?
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5.
2) 16.4 m
3) 4851 cm3
4) 5544 cm3
The radius of a circular cylinder is the same as that of sphere. Their volumes are
8.
4) Equal to its diameter
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3) Equal to its radius
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equal. The height of the cylinder is ?
4
2
1) 3 times its radius
2) 3 times its radius
The length of the wire of 0.2 mm readius that can be drawn after melting a solid
copper sphere of diameter 18 cm, is ?
2) 243 m
3) 2430 m
4) 24300 m
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1) 24.3 m
The radius of two cylinders are in the ratio of 2:3 and their heights are in the ratio 5:3.
.s
9.
The ratio of their volumes is ?
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1) 27:20
2) 20:27
3) 4:9
4) 9:4
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10. If the radius of a sphere is doubled, then its volume is increased by ?
1) 100%
2) 200%
3) 700%
4) 800%
11. The area of the base of a right circular cone is 154 cm2 and its height is 14 cm. The
curved surface of the cone is ?
(
1) 154 × 5
) cm2
2) 11 cm2
(
)
3) 154 + 7 cm2
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4) 5324 cm2
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10. The material of a cone is converted into the sphere of a cylinder of equal radius. If the
height of the cylinder is 5 cm, the height of the cone is ?
1) 10 cm
2) 15 cm
3) 18 cm
4) 24 cm
2. 3
3. 2
4. 4
5. 4
6. 3
7. 1
8. 4
9. 2
11. 1 12. 2
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du
c
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1. 2
m
ANSWERS
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