Volume of a Sphere Formula

Volume of a Sphere Formula

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

πr2h = Volume of a sphere + 1/3 × πr2h

Volume of a sphere = πr2h – 1/3 × πr2h

Volume of a sphere = 2/3 × πr2h

[For a sphere, h = d = 2r]

Volume of a sphere = 2/3 × πr2(2r)

Volume of a sphere = 4/3 × πr3

 

Volume of Sphere Formula

Volume of a sphere = 4/3 × πr3

 

Volume of Hollow Sphere Formula

Volume of a hollow sphere = 4/3 × π(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 × πr3

= 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 × πr3

= 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 × πr3

= 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.)

 

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