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Volume of a Sphere

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Volume of a Sphere is the amount of liquid a sphere can hold. 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. Volume of Sphere formula is given as 4/3πr3.

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.

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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

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 categeorsed 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

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

Derivation of Volume of 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

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 = \int_{y=-R}^{y=R} dV

\int_{y=-R}^{y=R}π(R^2 - y^2)dy

\pi|(R^2y - \frac{y^3}{3})dy|_{y=-R}^{y=R}

\pi \left[R^3-\frac{R^3}{3}-(-R^3+\frac{R^3}{3})\right]

\pi \left[2R^3-\frac{2R^3}{3}\right]

\frac{4}{3}\pi R^3

Thus, the formula for volume of sphere is derived.

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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

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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

= (4/3) (3.14) (9) (9) (9)

= (4) (3.14) (3) (9) (9)

= 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

= (4/3) (3.14) (12) (12) (12)

= (4) (3.14) (4) (12) (12)

= 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

= (4/3) (3.14) (6) (6) (6)

= (4) (3.14) (2) (6) (6)

= 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

= (4/3) (3.14) (4) (4) (4)

= (1.33) (3.14) (4) (4) (4)

= 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

= (4/3) (3.14) (5) (5) (5)

= (1.33) (3.14) (5) (5) (5)

= 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

= (4/3) (3.14) (8) (8) (8)

= (1.33) (3.14) (8) (8) (8)

= 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

= (4/3) (3.14) (7) (7) (7)

= (1.33) (3.14) (7) (7) (7)

= 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 sqaure of area 144 m2.

Volume of Sphere-FAQs

1. What is Volume of Sphere?

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

2. What is the Surface Area of a Sphere Formula?

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

3. What is the Formula for the Volume of a Sphere?

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

4. How do we find the Volume of the HemiSphere?

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

5. 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

6. 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.

7. 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.



Last Updated : 24 Jan, 2024
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