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.

What is Volume of a Sphere?

Volume of a sphere is the amount of space it takes up within it. The sphere is a three-dimensional round solid shape in which all points on its surface are equally spaced from its center. The fixed distance is the sphere’s radius, and the fixed point is the sphere’s center. We will notice a change in form when the circle is turned. As a result of the rotation of the two-dimensional object known as a circle, the three-dimensional shape of a sphere is obtained.

Learn More,

• Sphere
• Surface Area of Sphere

Volume of a Sphere Definition

Volume of a sphere is the total mass enclosed by the surface of the sphere. It is the 3-D space inside the sphere. It depends on the radius of the sphere. The image added below shows a sphere of radius “r” and its volume.

Volume of Sphere Formula

Volume of Sphere Formula is the formula that is used to find the volume of the sphere when its Radius is given. The volume of sphere formula for the sphere of radius R is added below,

Volume of Sphere Formula = 4/3πr³

Where,

• r is the Radius of a Sphere
• π is a Constant and its value is 22/7

A sphere is generally categorized into two that are,

• Volume of Solid Sphere
• Volume of Hollow Sphere

Let’s learn about them in detail.

Volume of a Solid Sphere

A solid sphere is a sphere which is completely filled till inside. i.e., it has mass till its core and its formula for the volume when its radius is “r” is,

Volume of a Solid Sphere(V) = (4/3)πr³

Volume of a Hollow Sphere

For a hollow sphere its internal space is empty and suppose its outer radius is R and its inner radius is r, then its volume is calculated using the formula,

Volume of Hollow Sphere = (4/3)π(R³ – r³)

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

How to Calculate Volume of Sphere?

Volume of sphere is the space occupied by a sphere. Its volume can be calculated using the formula  V = 4/3πr³.

Steps required to calculate the volume of a sphere are:

Stpe 1: Mark the value of the radius of the sphere.

Setp 2: Find the cube of the radius.

Step 3: Multiply the cube of the radius by (4/3)π

Step 4: Add the (unit)³ to the final answer.

Example to Calculate Volume of Sphere

Example: Find the volume of a sphere with a radius of 7 cm.

Given, r = 7 cm

V = (4/3)πr³

Volume of sphere, V = ((4/3) × π × 7³) cm³

V = 1436.8 cm³ 

Thus, the volume of sphere is 1436.8 cm³

Read More

• Volume of Cone
• Volume of Cube
• Volume of Cylinder

Volume of Sphere Examples

Example 1. Find the volume of the sphere whose radius is 9 cm.

Solution:

We have, r = 9

Volume of sphere = 4/3 πr³

⇒ Volume of sphere = (4/3) (3.14) (9) (9) (9)

⇒ Volume of sphere = (4) (3.14) (3) (9) (9)

⇒ Volume of sphere = 3052 cm³

Example 2. Find the volume of the sphere whose radius is 12 cm.

Solution:

We have, r = 12

Volume of sphere = 4/3 πr³

⇒ Volume of sphere = (4/3) (3.14) (12) (12) (12)

⇒ Volume of sphere = (4) (3.14) (4) (12) (12)

⇒ Volume of sphere = 7234.56 cm³

Example 3. Find the volume of the sphere whose radius is 6 cm.

Solution:

We have, r = 6

Volume of sphere = 4/3 πr³

⇒ Volume of sphere = (4/3) (3.14) (6) (6) (6)

⇒ Volume of sphere = (4) (3.14) (2) (6) (6)

⇒ Volume of sphere = 904.32 cm³

Example 4. Find the volume of the sphere whose radius is 4 cm.

Solution:

We have, r = 4

Volume of sphere = 4/3 πr3

⇒ Volume of sphere = (4/3) (3.14) (4) (4) (4)

⇒ Volume of sphere = (1.33) (3.14) (4) (4) (4)

⇒ Volume of sphere = 267.27 cm³

Example 5. Find the volume of the sphere whose diameter is 10 cm.

Solution:

We have, 2r = 10

⇒ r = 5

Volume of Sphere = 4/3 πr³

⇒ Volume of sphere = (4/3) (3.14) (5) (5) (5)

⇒ Volume of sphere = (1.33) (3.14) (5) (5) (5)

⇒ Volume of sphere = 522.025 cm³

Example 6. Find the volume of the sphere whose diameter is 16 cm.

Solution:

We have, 2r = 16

⇒ r = 8

Volume of sphere = 4/3 πr³

⇒ Volume of sphere = (4/3) (3.14) (8) (8) (8)

⇒ Volume of sphere = (1.33) (3.14) (8) (8) (8)

⇒ Volume of sphere = 2138.21 cm³

Example 7. Find the volume of the sphere whose diameter is 14 cm.

Solution:

We have, 2r = 14

⇒ r = 7

Volume of sphere = 4/3 πr³

⇒ Volume of sphere = (4/3) (3.14) (7) (7) (7)

⇒ Volume of sphere = (1.33) (3.14) (7) (7) (7)

⇒ Volume of sphere = 1432.43 cm³

Volume of Sphere-Practice Questions

Q1: Find the volume of the sphere whose diameter is 34 cm.

Q2: Find the volume of the hollow sphere whose inner is 4 cm and outer radius is 8 cm.

Q3: Find the volume of the sphere whose radius is 14 cm.

Q4: What is the volume of sphere whose radius is equal to the side of square with area 144 m².

Volume of Sphere-FAQs

What is Volume of Sphere?

Volume of Sphere is the space occupied by the surface of the sphere.

What is the Surface Area of a Sphere Formula?

Total Surface Area of Sphere of radius “r” is, Area = 4πr²

What is the Formula for the Volume of a Sphere?

Volume of a Sphere of radius “r” is, Volume = 4/3πr³

How do we find the Volume of the Hemisphere?

Volume of a Hemisphere of radius “r” is, Volume = 2/3πr³

What is the Ratio of Volume of Sphere and Hemisphere?

If a sphere and a hemisphere have the same radii then the ratio of their volume is,

V₁ : V₂ =  (4/3πr³) : (2/3πr³) = 2 : 1

What is the Unit of Volume of a Sphere?

Volume of the Sphere is measured in m³, cm³, litres, etc. m³ is the standard unit of measurement.

What is Volume of Sphere when its radius is Halved?

Volume of sphere = (4/3)πr³ = (4/3)π(r/2)³ = (4/3)π(r³/8) = Volume/8. So the volume of sphere gets one-eighth.