Volume of a Sphere
A sphere is a three-dimensional (3D)
figure. It has a face but no vertex and no edge. A football, glove, etc. are
the examples of the sphere. All the
points on the surface of a sphere are equidistant from a fixed point. This
fixed point is called its centre. The distance from the surface to the
centre of the sphere is called the radius of the sphere.
The space occupied by the sphere is
called its volume. If we draw
a circle on a sheet of paper, take a circular
disc, paste a string along its diameter and rotate it along the string. Then we find the shape
of a sphere.
The
unit of volume of a sphere is cubic unit. It means the unit of volume of a
sphere is cu m, cu cm, cu km, cu mm, etc.
We have
two types of spheres: solid sphere and hollow sphere.
We will learn to calculate the volume
of solid and hollow spheres.
Derivation of Volume of Sphere Formula
We can find the volume of a sphere
using the Archimedes’ principle.
Archimedes’ principle states that
“When an object sinks in water completely, it displaces water equal to its
volume”.
To find the volume of a sphere, let us
perform an activity.
Activity: Take a cylindrical flask full of water. Take a
cone and a sphere with the same radius and the same height as cylinder.
Now, put the cone and sphere inside the cylinder one by one so that they sink completely and water flows out. Collect the water flown out in a separate measuring flask. And measure the water of the measuring flask. You will observe the following:
|
Volume of water in cylindrical flask |
Volume of water flown out when cone is sunk |
Volume of water flown out when sphere is
sunk |
|
900 L |
300 L |
600 L |
From the above activity, we can conclude that the volume of a cylinder, a sphere and a cone with the same radii and the same height are in the ratio: 900 : 600 : 300, i.e., in the ratio 3 : 2 : 1.
So, the volume of a cylinder = volume
of a sphere + volume of a cone
Ï€r2h = Volume of a sphere +
1/3 × Ï€r2h
Volume of a sphere = Ï€r2h –
1/3 × Ï€r2h
Volume of a sphere = 2/3 × Ï€r2h
[For a sphere, h = d = 2r]
Volume of a sphere = 2/3 × Ï€r2(2r)
Volume of a sphere = 4/3 ×
Ï€r3
Volume of Sphere Formula
Volume of a sphere = 4/3 × Ï€r3
Volume of Hollow Sphere Formula
Volume of a hollow sphere =
4/3 × Ï€(r13 –
r23)
Volume of Sphere Examples
Example 1: Find the volume of a spherical balloon whose
radius is 7 m.
Solution: Given: Radius (r) = 7 m
Volume of a spherical balloon = 4/3 ×
Ï€r3
=
4/3 × 22/7 × 7 × 7 × 7
= 4/3 × 22 × 7 × 7
= 1437.33 cu m.
Example 2: Find the volume of a sphere whose diameter is
42 cm.
Solution: Given: Diameter (d) = 42 cm; Radius = 42/2 =
21 cm
Volume of a sphere = 4/3 × Ï€r3
= 4/3 × 22/7 × 21 × 21 × 21
= 4 × 22 × 21 × 21
= 38,808 cu cm.
Example 3: How many spherical marbles of radius 3.5 cm
can be made from a cylindrical iron rod whose base radius is 28 cm and height
is 1 m.
Solution: Given: Radius of spherical marble = 3.5 cm,
radius of cylindrical rod = 28 cm and height of rod = 1 m = 100 cm
Volume of a spherical marble = 4/3 ×
Ï€r3
= 4/3 × 22/7 × 3.5 × 3.5 × 3.5
= 3773/21 cu cm
Volume of a cylindrical iron rod = πr3h
= 22/7 × 28 × 28 × 100
= 1724800/7 cu cm
Number of marbles made = 1724800/7 ÷
3773/21 = 1371 (approx.)