Volume Of Sphere

The volume of sphere is the capacity it has. The shape of the sphere is round and three -dimensional. It has three axes such 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 radius of the sphere since if we take the cross-section of the sphere, it is a circle. The surface area of sphere 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 sphere is a three-dimensional shape which is the reason that we can measure Volume and area of a Sphere.

Sphere Volume

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 three-dimensional 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.

Also, read:

Volume of Sphere Formula with its Derivation

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

Volume of Sphere Derivation

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:

\(V = \int_{y=-R}^{y++R}dV\)

\(V = \int_{y=-R}^{y++R}\pi(R^{2}-y^{2})dy\)

\(V = \pi[R^{2}y – \frac{y^{3}}{3}]_{y=-R}^{y=+R}\)

Now, substitute the limits:

\(V = \pi[(R^{3}-\frac{R^{3}}{3})-(-R^{3}+\frac{R^{3}}{3})]\)

Simplify the above expression, we get:

\(V = \pi[2R^{3}-\frac{2R^{3}}{3}]\)

\(V =\frac{\pi}{3}[6R^{3}-2R^{3}]\)

\(V =\frac{\pi}{3}(4R^{3})\)

Thus, the volume of the sphere is \(V =\frac{4}{3} \pi R^{3}\) cubic units

Solved Examples

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. 

Stay tuned with BYJU’S – The Learning App for more information on volume of the three-dimensional objects and also learn other maths-related articles.

Frequently Asked Questions on Volume of Sphere

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 r^3
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^2
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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