Volume
Important Formula :
Cuboid :
Let Length = l
Breadth = b
Height = h then
i) Volume of Cuboid = (l X b X h ) cube units.
ii) Whole Surface Area of Cuboid
= 2(lb X bh X hl ) sq. units
iii) Length of Diagonal of Cuboid = √ l2+ b2+ h2 units.
Cube:
Let each edge of Cube be a units . Then,
i) Volume of the Cube = a3cubic units.
ii) Whole surface of the Cube = 6a2 sq. units.
iii) Diagonal of the Cube = √3a units.
Cylinder :
Let radius of base be r & height or length be h .
i) Volume of the Cylinder = πr2h cubic units.
ii) Curved surface area of the Cylinder = 2πrh sq. units
iii) Total surface area of the Cylinder = 2πr2 + 2πrh
= 2πr( r + h )
Cone :
Let radius of base = r , height = h & slant height = l.
i) l = √ h2+ r2
1
ii) Volume of the Cone = -------πr2h sq. units.
3
iii) Curved surface area of the Cone = πrl sq. units.
iv) Total surface area of the Cone = πr2+ πrl
= πr(r + l) sq. units.
Sphere :
Let radius of Sphere be r. Then
4
i) Volume of Sphere =--------- πr3 cubic units
3
ii) Surface area of the Sphere =. 4πr2 sq. units
2
iii) Volume of a hemisphere = -------πr3 cubic units
3
iv) Curved surface area of the hemisphere = 2πr2 sq. units
v) Total Surface area of the hemisphere = 3πr2 sq. units
Prism :
i) Volume of a Prism = Area of base X height
Example :
1) Find the volume and surface area of a sphere of diameter 21 cm .
Solution :
4
Volume of Sphere =--------- πr3 cubic units
3
4 22 21 21 21
Vol.of Sphere =-----X-----X ----X------X------ cubic units
3 7 2 2 2
4 22
= -------X------X10.5X10.5X10.5cubic units
3 7
= 4851 cubic units
Surface area of the Sphere =. 4πr2 sq. units
22 21 21
S. ar. of Sphere =. 4 X -----X-----X------ sq. cm.
7 2 2
=1386 sq. cm.
2) Find the volume, curved surface area and total surface area of a hemisphere of diameter 21 cm.
Solution :
2
Volume of a hemisphere = -------πr3 cubic units
3
2 22 21 21 21
= -------X------X-------X-------X------- cubic units
3 7 2 2 2
=22425.5 Cubic cm.
Curved surface area of the hemisphere = 2πr2 sq. units
22 21 21
= 2 X-------X-------X-------- sq. cm.
7 2 2
= 693 sq. cm.
Total Surface area of the hemisphere = 3πr2 sq. units
= 3(22/7)(10.5)2 sq. cm.
=1039.5 sq. cm.
3) How many bullets can be made out of lead cylinder 14 cm high and 6 cm radius ,each bullet being 1.5 cm in diameter.
Solution :
Volume of cylinder in cubic cm.
No. of bullets =-----------------------------------------
Volume of 1 bullet in cubic cm.
π X 6 X 6 X 14
= -----------------------------------------
(4/3) X π X 0.75 X 0.75 X 0.75
= 896
4) Find the no. of lead balls of diameter 1 cm each that can be made from a sphere of diameter 16 cm.
Solution :
Volume of big sphere
No. of lead balls =----------------------------------
Volume of 1 small ball
4
-------- π X 8 X 8 X 8
3
= ------------------------------------
4
--------- π X 0.5 X 0.5 X 0.5
3
= 4096
5) Find the area of the metal sheet required to prepare a cone 24 cm high with base radius 7 cm.
Solution :
l = √ (24)2 +(7 )2
= 25 cm.
Ar. of the sheet = Total S. Ar. of Cone
= πr(r + l) sq. units.
= ( 22/7 )X 7 X ( 25 + 7 )
= 704 sq. cm.
Exercise
1) Which of the following pairs is not correctly matched:
Geometrical objects No. of Vertices
a) Tetrahedron 4
b) Pyramid with rectangular base 5
c) Cube 6
d) Triangle 3
2) If the length, breadth and height of a cuboid are 2 m, 2 m and 1 m respectively, then its surface area ( in sq. m ) is :
a) 8 b) 12 c) 16 d) 24
3) A beam 9 m long, 40 cm wide and 20 cm deep is made up of iron which weight 50 kg per cubic metre. The weight of the beam is :
a) 27 kg b) 36 kg c) 48 kg d) 56 kg
4) The length of the longest rod that can be placed in a room 30 m long, 24 m broad and 18 m high, is
a) 30 m b) 15 √7 m c) 60 m d) 30 √2 m
5) If 1 cm3 of silver weight 10 gms and 8 gms of silver costs Rs. 30, then the cost of a silver cube of edge 4 cm, is :
a) ₹ 800 b) ₹ 1200 c) ₹ 1920 d) ₹ 2400
6) The volume of a cube with surface area 384 sq. cm, is :
a) 216 cm3 b) 256 cm3 c) 484 cm3 d) 512 cm3
7) If the length of diagonal of a cube is 4√3 cm, then length of its edge is :
a) 2 cm b) 3 cm c) 4 cm d) 6 cm
8) If the areas of 3 adjacent sides of a cuboid are x , y , z respectively, then the volume of the cuboid is :
a) xyz b) 2xyz c) √xyz d) 3√xyz
9) The length , breadth and height of a cuboid are in the ratio 1:2:3 . The length and height of the cuboid are increased by 100%, 200% and 200% respectively. Then, the increase in the volume of the cuboid is :
a) 5 times b) 6 times c) 12 times d) 17 times
10) The volume of the cylinder whose height is 14 cm and diameter of base 4 cm, is :
a) 176 cm3 b) 196 cm3 c) 276 cm3 d) 352 cm3
Answer
1) c 2) c 3) b 4) d 5) d 6) d 7) c 8) c 9) d 10) a
Comments