Volume Of A Sphere- Definition, Derivation, Formula

Safalta expert Published by: Yashaswi More Updated Wed, 23 Feb 2022 10:54 AM IST

Highlights

Check out how to find the volume of a sphere- definition and the derivation of formula here at Safalta.com

The volume of sphere is the capacity it has. It is the space occupied by the sphere. The volume of sphere is measured in cubic units, such as m3, cm3, in3, etc. The shape of the sphere is round and three-dimensional. It has three axes as x-axis, y-axis and z-axis which defines its shape.

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All the things like football and basketball are examples of the sphere which have volume.
The volume here depends on the diameter of the radius of the sphere since if we take the cross-section of the sphere, it is a circle. The surface area of r is the area or region of its outer surface. To calculate the sphere volume, whose radius is ‘r’ we have the below formula:

 

Volume of a sphere = 4/3 πr3
 

Now let us learn here to derive this formula and also solve some questions with us to master the concept.

If you consider a circle and a sphere, both are round. The difference between the two shapes is that a circle is a two-dimensional shape and a sphere is a three-dimensional shape which is the reason that we can measure the Volume and area of a Sphere.


 

What is the Volume of Sphere?

The volume of sphere is the amount of space occupied, within the sphere. The sphere is defined as the three-dimensional round solid figure in which every point on its surface is equidistant from its centre. The fixed distance is called the radius of the sphere and the fixed point is called the centre of the sphere. When the circle is rotated, we will observe the change of shape. Thus, the 3D Shape sphere is obtained from the rotation of the two-dimensional object called a circle.

Archimedes’ principle helps us find the volume of a spherical object. It states that when a solid object is engaged in a container filled with water, the volume of the solid object can be obtained.

Because the volume of water that flows from the container is equal to the volume of the spherical object.


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Volume of Sphere Formula with its Derivation

The formula to find the volume of sphere is given by:

Volume of sphere = 4/3 πr3   [Cubic units]

Let us see how to derive the dimensional formula for the volume of a sphere.

 

Derivation:

The volume of a Sphere can be easily obtained using the integration method.

Assume that the volume of the sphere is made up of numerous thin circular disks which are arranged one over the other as shown in the figure given above. The circular disks have continuously varying diameters which are placed with the centres collinearly. Now, choose any one of the disks. A thin disk has radius “r” and the thickness “dy” which is located at a distance of y from the x-axis. Thus, the volume can be written as the product of the area of the circle and its thickness dy. 

Also, the radius of the circular disc “r” can be expressed in terms of the vertical dimension (y) using the Pythagoras theorem. 

Thus, the volume of the disc element, dV can be expressed by:

dV =(πr2)dy

dV =π (R2-y2) dy

Thus, the total volume of the sphere can be given by:

V=∫y=+Ry=−RdVV=∫y=−Ry=+RdV

V=∫y=+Ry=−Rπ(R2−y2)dyV=∫y=−Ry=+Rπ(R2−y2)dy

V=π[R2y–y33]y=+Ry=−RV=π[R2y–y33]y=−Ry=+R

Now, substitute the limits:

V=π[(R3−R33)−(−R3+R33)]V=π[(R3−R33)−(−R3+R33)]

Simplify the above expression, we get:

V=π[2R3−2R33]V=π[2R3−2R33]

V=π3[6R3−2R3]V=π3[6R3−2R3]

V=π3(4R3)V=π3(4R3)

Thus, the dimensional formula of volume of the sphere is V=4/3πR3V=4/3πR3 cubic units. 

 




 

What is the formula for volume of sphere?

The formula to calculate the volume of sphere is defined by: 4/3rd of Pi and cube of radius of sphere.

How to calculate volume of a sphere?

To find the volume of sphere we have to use the formula:
Volume = 4/3 π r3
Where ‘r‘ is the radius of the sphere.

What is the total surface area of the sphere?

The total surface area of any given sphere is equal to;
A = 4πr square where ‘r’ is the radius of the given sphere.

What is the ratio of volume of sphere and volume of cylinder?

The volume of any sphere is 2/3rd of the volume of any cylinder with equivalent radius and height equal to the diameter.

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