Volume Of A Sphere- Definition, Derivation, Formula

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. 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:

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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. 

 

 

How to Calculate Volume of Sphere?

The volume of sphere is the space occupied within it. It can be calculated using the above formula, which we have already derived. To find the volume of a given sphere follow the steps below:

• Check with the radius of the given sphere. If the diameter of the sphere is known, then divide it by 2, to get the radius
• Find the cube of the radius r3
• Now multiply it with (4/3)π
• The final answer will be the volume of sphere

Let us see some examples of calculating the volume of spheres of different dimensions.


 

Solved Examples on Volume of Sphere

Q.1: Find the volume of a sphere whose radius is 3 cm?

Solution :

Given: Radius, r = cm

Volume of a sphere = 4/3 πr3 cubic units

V = 4/3 x 3.14 x 33

V = 4/3 x 3.14 x 3 x 3 x 3

V = 113.04 cm3
 

Q.2: Find the volume of sphere whose diameter is 10 cm.

Solution: Given, diameter = 10 cm

So, radius = diameter/2 = 10/2 = 5 cm

As per the formula of sphere volume, we know;

Volume = 4/3 πr3 cubic units

V = 4/3 π 53

V = 4/3 x 22/7 x 5 x 5 x 5

V = 4/3 x 22/7 x 125

V = 523.8 cu.cm.