Volume of Sphere Formula Derivation

Volume of sphere formula can be derived using the following methods:

• Using Integration
• Using Archimedes Relationship between Cylinder, Cone and Sphere

Let’s discuss these methods in detail as follows:

Volume of Sphere Using Integration

Using the integration approach, we can simply calculate the volume of a sphere.

Suppose the sphere’s volume is made up of a series of thin circular discs stacked one on top of the other, as drawn in the diagram above. Each thin disc has a radius of r and a thickness of dy that is y distance from the x-axis.

Let the volume of a disc be dV. The value of dV is given by,

dV = (πr²)dy

Thus, dV = π (R² – y²)dy

The total volume of the sphere will be the sum of volumes of all these small discs. The required value can be obtained by integrating the expression from limit -R to R.

So, the volume of sphere becomes,

V = [Tex]\int_{y=-R}^{y=R} dV [/Tex]

⇒ V = [Tex]\int_{y=-R}^{y=R}π(R^2 – y^2)dy [/Tex]

⇒ V = [Tex]\pi|(R^2y – \frac{y^3}{3})dy|_{y=-R}^{y=R} [/Tex]

⇒ V = [Tex]\pi \left[R^3-\frac{R^3}{3}-(-R^3+\frac{R^3}{3})\right] [/Tex]

⇒ V = [Tex]\pi \left[2R^3-\frac{2R^3}{3}\right] [/Tex]

⇒ V = [Tex]\frac{4}{3}\pi R^3 [/Tex]

Thus, the formula for volume of sphere is derived.

Volume of Sphere Using Archimedes Relations

As Archimedes has already proved, if a cone, a sphere, and a cylinder have the same radius r and the same height, their volumes are in the ratio of 1:2:3.

Therefore we can say:

 Volume of Cylinder = Volume of Cone + Volume of Sphere

Thus, Volume of Sphere = Volume of Cylinder – Volume of Cone

As we know, that volume of cylinder = πr²h and volume of cone = (1/3)πr²h

Substituting these values into the equation, we get:

Volume of Sphere = πr²h – (1/3)πr²h = (2/3)πr²h

We assume that the height of the cylinder equals the diameter of the sphere, which is 2r. Thus:

Volume of sphere is (2/3)πr²h = (2/3)πr²(2r) = (4/3)πr³

Also, Check

• Spherical Cap Volume Formula
• Spherical Sector Formula
• Spherical Segment Formula

Volume of a Sphere is the amount of liquid a sphere can hold. Volume of Sphere formula is given as 4/3πr³. It is the space occupied by a sphere in 3-dimensional space. It is measured in unit³ i.e. m³, cm³, etc. A sphere is a three-dimensional solid object with a round form in geometry.

The volume of the sphere is the total space occupied by the surface of the sphere and it is proportional to the cube of the radius of the sphere. In this article, we will learn about Volume of Sphere, Volume of Sphere Formula, Volume of Sphere Formula Examples, and others in detail.