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So, let’s start the learning now.
Getting Started
This post contains three sections –
- Section A – This section contains important formulas, short tricks, concepts on volume of solids.
- Section B – This section contains some examples that demonstrates how to solve questions on volume of solids using short tricks, formulas shown in section A.
- Section C – Go to this section to solve some problems on volume of solids.
Now,
Where to start from?
We recommend go through section A, then section B, then section C. Even if you are familiar with this topic, do not just skip section A and section B.
Section A: (Terms & Formulas)
Volume: The Volume of a solid is the number of unit cubes it takes to fill up a solid. There are different types of solids for which volume can be found out.
These are mentioned below:
Shape | Volume |
Cube | Volume of Cube = s^{3} Where, s = Length of the side Example:Find the volume of cube whose side is 3 cm. Solution: |
Cuboid |
Volume of cuboid = l * w * h
Where, Example: Solution: |
Prism |
Volume of Prism = (Area of base) * height
Where, Area of base depends on type of prism. For rectangular prism, For Triangular prism, Example:Find the volume of triangular prism given below – Solution: Volume(v) of the prism = (Area of base) * height |
Cylinder |
Volume of cylinder = π * r² * h
Where, Example:Find the volume of cylinder of radius 3 cm and height 5 cm. Solution: Volume of cylinder = π * r² * h, So, V = π * 3² * 5, |
Hollow Cylinder |
Volume of Hollow cylinder = π * h * (R² – r²) Where, R = Radius of the outer cylindrical surface, r = Radius of the inner cylindrical surface, h = Height of the hollow cylinder Example:Find the amount(in cm^{3}) of metal used in making a hollow cylinder having thickness 2 cm, outer radius 14 cm and height 70 cm. Solution:The thickness of the hollow metallic cylinder = 2 cm. |
Cone |
Volume of cone = (1 / 3) * π * r² * h Where, r = Radius of circular base, h = Height of cone Example:Find the volume of cone having radius 3 cm and height 11 cm. Solution: |
Pyramid |
Volume of pyramid = (1 / 3) * B * h
Where, Example:Find the volume of pyramid given below – Solution: Where, Given, |
Sphere |
Volume of Sphere = (4 / 3) * π * r³ Where, r = Radius of sphere Example:Find the volume of sphere having radius 7 inch. Solution: |
Hemisphere |
Volume of Hemisphere = (2 / 3) * π * r³ Where, r = Radius of Hemisphere Example:Find the volume of hemisphere having radius 15 inch. Solution: As we know, So, |
Section B: (Examples)
Now, we will see how to approach any questions on volume of solids. What are the key factors we should look into before trying to solve such questions. Please do not move to next section without going through each volume of solids questions and answers.
Example 1
A rectangular prism has a volume of 4200 mm^{3} . It has a rectangular base of length 10 mm and width 6 mm. Find the height of the prism.
Solution
Below image clearly depicts the scenario,
Given,
Volume of prism = 4200 mm^{3} ,
Length of prism = 10 mm,
Width of prism = 6 mm
Volume of prism = (Area of Base * height)
Area of Base = Length * Width
So,
=> 4200 = (l * w) * h
=> 4200 = (10 * 6) * h
=> 4200 = 60 * h
h = 70 mm
Example 2
The length of a side of a cube is equal to 8 cm. Find the Volume of the cube.
Solution
Volume(v) of cube = s^{3 }Where, s is side of a cube
Side(s) of cube = 8 cm
So, volume = 8^{3}
= 8 * 8 * 8
Volume = 512 cm^{3}
Example 3
Find the thickness of the hollow cylinder of height 100 cm if the volume between the inner and outer cylinders is 13000π mm^{3} and outer diameter is 14 mm.
Solution
At first, let’s convert dimensions into common unit. So, we will convert it into mm unit (10 mm. = 1 cm.)
Height of cylinder = 100 cm = 1000 mm
Outer diameter (d) = 14 mm
So, Outer radius (R) = 7 mm (Diameter = 2 x Radius)
Below diagram depicts the scenario.
Here,
r = Inner radius of cylinder,
R = Outer radius of cylinder,
h = Height of cylinder
So, we have to find R – r .
Volume (V) of Cylinder = 13000π mm^{3}
V = π * h * ( R^{2} – r^{2} )
13000π = π * 1000 * (7^{2} – r^{2} )
7^{2} – r^{2} = 13
49 – r^{2} = 13
r^{2} = 49 – 13
r^{2} = 36
r = 6 mm
So, Thickness of Cylinder = R – r
= 7 – 6
= 1 mm
Example 4
Find the volume of cone with radius 5 cm and height 10 cm.
Solution
Volume (V) of cone = (1 / 3) * π * r^{2} * h
Where,
r = radius of base,
h = height of code
Given,
r = 5 cm,
h = 10 cm,
π = 22 / 7
So, Volume = (1 / 3) * (22 / 7) * ( 5^{2} ) * 10
Volume of cone = 261.90 cm^{3}
Section C: (Exercises)
In this section, we will see different volume of solids questions and answers (with short tricks, if available).
Mark against the correct answer:
At first, please try to solve question by yourself, then move to see it’s answer.
Question 1
The radius of a sphere is 3r, then it’s volume will be –
a. (4 / 3)πr^{3}
b. 36πr^{3}
c. (8 / 3)πr^{3}
d. (32 / 3)πr^{3}
Question 2
The total surface area of a cube is 96 cm^{2}. The volume of cube is –
a. 16 cm^{3}
b. 512 cm^{3}
c. 64 cm^{3}
d. 27 cm^{3}
Question 3
A cube of side 4 cm contains a sphere touching its sides. Find the volume of gap in between.
a. 30.48 cm^{3}
b. 54 cm^{3}
c. 46 cm^{3}
d. 29 cm^{3}
Question 4
Find the amount of water displaced by a solid spherical ball of diameter 4.2 cm, when it is completely immersed in water.
a. 25.6 cm^{3}
b. 38.80 cm^{3}
c. 43 cm^{3}
d. 56.60 cm^{3}
Question 5
A shopkeeper has one jamun of radius 5 cm. With the same amount of material, how many jamun of radius 2.5 cm can be made?
a. 5
b. 10
c. 8
d. 6
Question 6
A right triangle with sides 6 cm, 8 cm, and 10 cm is revolved about the side 8 cm. Find the volume of the solid.
a. 330.6 cm^{3}
b. 185.9 cm^{3}
c. 200 cm^{3}
d. 301.7 cm^{3}
Question 7
A cylindrical tube which is opened at both the ends is made of iron sheet which is 2 cm thick. If the outer diameter is 16 cm and length is 100 cm, find how many cubic centi meters of iron has been used in making tube ?
a. 8800 cm³
b. 9000 cm³
c. 5000 cm³
d. 3000 cm³
Question 8
A steel vessel has a base of length 30 cm and breadth 15 cm. Water is poured in the vessel. A cubical steel box having edge of 15 cm is immersed in the vessel. How much will the water rise?
a. 7.5 cm
b. 10 cm
c. 15 cm
d. 30 cm
Question 9
A cylinder is having radius 1 m and height 5 m is completely filled with juice. In how many conical flasks can this juice be filled if the flask radius and height is 50 cm each?
a. 50
b. 600
c. 120
d. 90
Question 10
2 cm thick 15 metal plates are kept exactly one above the other. On top of the metal plate, a hemisphere of diameter 6 cm is kept. This just covers the top plate. What is the volume of the entire object?
a. 360π cm^{3}
b. 144π cm^{3}
c. 81 cm^{3}
d. 288π cm^{3}
Question 11
A metallic sheet is of rectangular shape with dimensions 48 m x 36 m. From each corner of the sheet, a square is cut off so as to make an open box. If the length of the square is 10 m, the volume of the box (in m^{3}) is:
a. 4480
b. 5120
c. 6420
d. 8960
Question 12
The surface of a cube is 24 cm^{2} . Find its volume ?
a. 8 cm^{3}
b. 16 cm^{3}
c. 32 cm^{3}
d. 40 cm^{3}
Question 13
The areas of the two spheres are in the ratio 1 : 16. The ratio of their volume is ?
a. 1 : 4
b. 1 : 16
c. 1 : 8
d. 1 : 64
Question 14
A cuboidal garage has a volume of 480 m³ a length of 10 m and a width of 8 m. What is the height of the garage?
a. 4 m
b. 5 m
c. 6 m
d. 7 m
Question 15
Which set of dimensions belongs to a right rectangular prism with a volume of 550 cm^{3} ?
a. 6 cm , 12 cm , 5 cm
b. 10 cm , 5 cm , 11 cm
c. 7 cm , 9 cm , 6 cm
d. 12 cm , 2 cm , 5 cm
Question 16
A standard dice is a cube having sides of 10 mm. What is the volume of the dice?
a. 30 mm³
b. 45 mm³
c. 425 mm³
d. 1000 mm³
Question 17
A cylindrical bowl has a radius of 4.3 cm and a height of 11.6 cm. What is the volume of the bowl to the nearest tenth of a cubic centimetre?
a. 49.9 cm³
b. 168.4 cm³
c. 673.5 cm³
d. 1816.8 cm³
Question 18
A metallic rod has a diameter of 17 cm and has 72 cm between the two ends. What is the volume of the space between the ends?
a. 10300.4 cm^{3 }
b. 3589.5 cm^{3}
c. 67890.4 cm^{3}
d. 16334.2 cm^{3}
Question 19
A hockey puck has a diameter of 7.6 cm and a height of 3.4 cm. What is the volume of a cylindrical package containing 5 pucks?
a. 770.8 cm³
b. 480.7 cm³
c. 570.3 cm³
d. 690.4 cm³
Question 20
A swimming pool 9 m wide and 10 m long is 1 m deep on the shallow side and 5 m deep on the deeper side. Its volume is:
a. 390 m³
b. 270 m³
c. 420 m³
d. 530 m³
Question 21
A circular well with a diameter of 2 meters, is dug to a depth of 21 meters. What is the volume of the earth dug out?
a. 46 m^{3}
b. 79 m^{3}
c. 66 m^{3}
d. 28 m^{3}
Question 22
Two right circular cylinders of equal volumes have their heights in the ratio 1 : 4. Find the ratio of their radii.
a. 2 : 1
b. 4 : 1
c. 1 : 3
d. 1 : 2
Question 23
The radius of the base of a cone is 4 cm and its volume is 68.32 cm³. (π = 3.14) What is the height of the cone?
a. 16.34 cm
b. 28.48 cm
c. 47.39 cm
d. 12.48 cm
Conclusion
We have seen how to find volume of solids like cube, cuboid, prism, cylinder, hollow cylinder, cone, sphere and hemisphere.
List of important formulas for volume of solids –
Shape | Volume |
Cube | Volume of Cube = s^{3} Where, s = Length of the side |
Cuboid | Volume of cuboid = l * w * h
Where, |
Prism | Volume of Prism = (Area of base) * height
Where, Area of base depends on type of prism. For rectangular prism, For Triangular prism, |
Cylinder | Volume of cylinder = π * r² * h
Where, |
Hollow Cylinder | Volume of Hollow cylinder = π * h * (R² – r²) Where, R = Radius of the outer cylindrical surface, r = Radius of the inner cylindrical surface, h = Height of the hollow cylinder |
Cone | Volume of cone = (1 / 3) * π * r² * h Where, r = Radius of circular base, h = Height of cone |
Pyramid | Volume of pyramid = (1 / 3) * B * h Where, B = Area of base, h = height of pyramid |
Sphere | Volume of Sphere = (4 / 3) * π * r³ Where, r = Radius of sphere |
Hemisphere |
Volume of Hemisphere = (2 / 3) * π * r³ Where, r = Radius of Hemisphere |