Volume of a Sphere: Formula, Derivation & Solved Examples

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Volume of a Sphere is referred to as the measurement of the space that it occupies. The volume of a sphere is measured in cubic units, such as m³, cm³, in³, etc. The shape of the sphere is round and three-dimensional that has no edges or vertices. It has three axes which are, the x-axis, y-axis, and z-axis which define its shape. Here, we will be learning to find the volume of a sphere by deducing the formula of the volume of a sphere.

Table of Content

What is Volume of Sphere?
Volume of a Sphere Formula
Derivation of the Volume of a Sphere
How to Calculate Volume of a Sphere?
Solved Examples on Volume of Sphere
Things to Remember
Sample Questions

Key Terms: Sphere, Volume of Sphere, Radius, Diameter, Volume, Three-dimensional Shape, Cone, Cylinder

What is Volume of Sphere?

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The sphere is a three-dimensional structure. Like a circle, the sphere also possesses a center and a radius. All the points at a distance of radius in space together form a sphere. Twice the distance of a radius is called its diameter.

Sphere

Sphere

Also Read: Edges, Faces, and Vertices

Volume of a Sphere Formula

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The formula for the volume of a sphere is calculated using the radius of the three-dimensional structure. Spheres can be of two types, solid and hollow. The formula which is used to calculate the volume of a solid sphere is given below:

Volume of Sphere = 4/3 πr³

Where, r is the Radius of the Sphere.

In the case of a hollow sphere, there is a thin layer that has mass or volume, and it makes the hollow sphere. So, the volume is calculated as the

Volume of Hollow Sphere = (Volume of Outer Sphere) – (Volume of Inner Sphere)

Volume of Hollow Sphere = 4/3 πR³ - 4/3 πr³

Where

R = Radius of Outer Sphere
r = Radius of the Inner Sphere

Volume of sphere needs the value of the radius or its diameter. In the case of diameter, the value has to be halved to get the radius of the sphere.

Also read: Sphere Formula

Derivation of the Volume of a Sphere

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The formula can be derived using two methods, the Arrhenius method, and by integration method. Both of the methods used to derive the formula are explained below:

Arrhenius Method

According to this method, the volume of a sphere is calculated using the volume of a cone and a cylinder that have an equal radius, cross-section area and also height equal to twice the radius. According to this method, the volume of such a cylinder is equal to the sum of the volume of a cone, and sphere, further these volumes are in the ratio 1:2:3 (cylinder: sphere: cone).

Thus,

Volume of a cylinder = Volume of sphere + Volume of a cone

Volume of sphere = Volume of cylinder – Volume of cone

Volume of cylinder = 3 x Volume of cone

Volume of a cylinder = πr²h …….. (1)

πr²h = 3 x Volume of cone

Volume of the cone = πr²h/3 …….. (2)

Volume of sphere = Volume of cylinder – Volume of cone

Volume of sphere = πr²h - πr²h/3

Volume of sphere = 2 πr²h/3

h = 2r

Volume of sphere = 2 πr² 2r/3

Volume of sphere = 4 πr³/3

The volume of a solid sphere = 4/3 π r³

Also Read: Quadrilateral

Integration Method

Integration Method of Volume of Sphere

Integration Method

Integration method assumes that the sphere is made up of circular discs having thickness dy and radius x. They are placed at a different height from the center of the sphere, have radius r, and have varied radiuses. The disc selected for the derivation is placed at the height z from the central plane.

According to Pythagoras' theorem:

x² + z² = r²

x = r² - z²

Area of the disk thus is π (r² – z²)

Now we need to calculate the area of all the discs that would make up the sphere, thus we apply integration to the formula above from +r to -r.

V = r∫ -r (r²–z²)dy

V = π (zr² – z²/3)

V = π [2/3r³ – (-2/3 r³)]

Thus, the volume of a solid sphere = 4/3 π r³

Also Read: Surface Areas and Volumes Revision Notes

How to Calculate Volume of a Sphere?

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

Check the value of the radius of the given sphere. If the diameter of the sphere is known, then divide it by 2, to get the radius
Then take the cube of the radius r³
Now multiply it with (4/3)π
Finally, add the units to the final answer.

Solved Examples on Volume of Sphere

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Example 1: Find the volume of a sphere having a radius of 5 cm.

Solution: Given, the volume of a sphere formula,

V = 4/3 πr³

= (4/3) x 3.14 x 5³

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

= 523.3 cm³

Example 2: Find the radius of the sphere when the volume of a sphere is 523 cubic yards.

Solution: Given,

V = 523 cubic yards

Volume of a sphere, V = 4/3 πr³

523 = (4/3) x 3.14 x r³

523 = 4.19r³

Divide both sides by 4.19

r³ = 124.82

³√r³ = ³√124.82

r = 5

So, the radius of the sphere is 5 yards.

Example 3: Find the volume of a sphere with a radius of 24 mm.

Solution: As we know the radius is half of the diameter, then

r = 24/2 = 12 mm

The volume of a sphere = 4/3 πr³

By substitution, we get

V = (4/3) x 3.14 x 12 x 12 x 12 = 7734.6 mm³

Example 4: The radius of an inflated spherical balloon is 7 feet. If air is leaking from the balloon at a constant rate of 26 cubic feet per minute, then how long will it take for the balloon to be completely deflated?

Solution: Given that

Volume of the spherical balloon = 4/3 πr³

Volume = 4/3 x 3.14 x 7 x 7 x 7 = 1436.03 cubic feet

Now, divide the volume of the balloon by the rate of leakage

Time in minutes = 1436.03 cubic feet/26 cubic feet = 55 minutes

Read More:

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Triangles	Scalene Triangle	Right-angled triangle
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Similarities of Triangle	Vertex	Vector Formula

Things to Remember

A sphere is a three-dimensional structure.
The properties of a sphere are similar to a circle.
The volume of a solid sphere = 4/3 πr³
The volume of hollow sphere = 4/3 πR³ - 4/3 πr³
The surface area of a sphere is 4 πr²
The formula to derive the formula to calculate the volume of a sphere can be of two ways: Arrhenius and integration method.

Read More: Perimeter of Parallelogram

Sample Questions

Ques. Find the volume of a ball having a radius of 6 m. (2 Marks)

Ans. Radius of the ball = 6 m

Volume = 4/3 πr³

V = 4 x 22 x 6x6x6/3 x 7

Volume = 905.14 m³

Ques. Find the volume of a sphere having a diameter of 10 m. (3 Marks)

Ans. Diameter = 10 m

Radius = Diameter/2 = 5 m

Volume = 4/3 πr³

V = 4 x 22 x 5x5x5/3 x 7

Volume = 523.8 m³

Ques. A spherical vessel has an inner diameter of 14 m, find its capacity. (3 Marks)

Ans. Capacity of the vessel = volume

Diameter = 14 m

Radius = 7 m

Volume = 4/3 πr³

V = 4 x 22 x 7x7x7/3 x 7

Volume = 1378.6 m³

Ques. A sphere is made up of a thick sheet having a thickness of 1 cm. The inner radius is 5 cm. Find the volume of metal used. (4 Marks)

Ans. Volume of metal used = volume of metal and hollow sphere.

The volume of hollow sphere = 4/3 πR³ - 4/3 πr³

R’ = outer radius = 5 + 1 = 6 cm

R” = inner radius = 5 cm

Volume of hollow sphere = 4/3 πR³ - 4/3 πr³

The volume of hollow sphere = 4/3 π (R³ - r³)

The volume of hollow sphere = 4/3 π (125 - 108)

The volume of hollow sphere = 71.2 cm³

Ques. A metal cube has a side of 7 cm. How many spheres can be made out of it having radius 1 cm. (2 Marks)

Ans. Side of cube = 7cm

Volume of cube = 441 cm³

Volume of sphere to be made = 4/3 πr³ = 4.18 cm³

Number of spheres that can be made = 441/4.18 cm³ = 105.5 spheres

Ques. A sphere has a volume of 1000 m3, what is its diameter? (2 Marks)

Ans. Volume = 1000 m3 = 4/3 πr³

r = 31000x3x7/4x22

Radius = 6.2 m

Ques. A cone and a sphere have equal volume and equal radius, find the ratio of the height of the cone and the radius of the sphere. (2 Marks)

Ans. Volume of sphere = volume of the cone

4/3 πr³ = 1/3 πr²h

r = h/4

h = 4r

Thus, the ratio of r: h is 4:1

Ques. A cylinder has a height equal to three times its radius. Its volume is equal to the volume of a sphere having a radius of 4 cm. Find the radius of the cylinder. (4 Marks)

Ans. Height of cylinder = 3 r’ (radius of cylinder)

Volume of cylinder = volume of sphere

πr’²h = 4/3 πr³

πr’²h = 4 x 22 x 4x4x4/3 x 7

πr’²h = 268.19

πr’² 3r’ = 268.19

3πr’³ = 268.19

r’ = 50.2 cm

Ques. Find the volume of a ball having a radius of 5 cm. (2 Marks)

Ans. Radius of the ball = 5 cm

Volume = 4/3 πr³

V = 4 x 22 x 5x5x5/3 x 7

Volume = 523.8 cm³

Ques. Find the volume of a sphere having a diameter of 20 cm. (2 Marks)

Ans. Diameter = 20 cm

Radius = Diameter/2 = 10 cm

Volume = 4/3 πr³

V = 4 x 22 x 10x10x10/3 x 7

Volume = 4190.4 cm³

Ques. Find the volume of a hemisphere ball having a radius of 7 cm. (2 Marks)

Ans. Radius of the ball = 7 m

Volume = 4/3 πr³

V = 4 x 22 x 7x7x7/3 x 7

Volume = 1378.6 m³

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Volume of a Sphere: Formula, Derivation & Solved Examples

What is Volume of Sphere?

Volume of a Sphere Formula

Derivation of the Volume of a Sphere

Arrhenius Method

Integration Method

How to Calculate Volume of a Sphere?

Solved Examples on Volume of Sphere

Things to Remember

Sample Questions

CBSE X Related Questions

1.
In a right triangle ABC, right-angled at A, if $\sin B = \dfrac{1}{4}$, then the value of $\sec B$ is

2.
Let $p$, $q$ and $r$ be three distinct prime numbers. Check whether $pqr + q$ is a composite number or not. Further, give an example for three distinct primes $p$, $q$, $r$ such that
(i) $pqr + 1$ is a composite number
(ii) $pqr + 1$ is a prime number

4.
In the adjoining figure, $\triangle CAB$ is a right triangle, right angled at A and $AD \perp BC$. Prove that $\triangle ADB \sim \triangle CDA$. Further, if $BC = 10$ cm and $CD = 2$ cm, find the length of AD.

5.
Prove that \(\dfrac{\sin \theta}{1 + \cos \theta} + \dfrac{1 + \cos \theta}{\sin \theta} = 2\csc \theta\)

6.
From one face of a solid cube of side 14 cm, the largest possible cone is carved out. Find the volume and surface area of the remaining solid.
Use $\pi = \dfrac{22}{7}, \sqrt{5} = 2.2$

Similar Mathematics Concepts

Comments

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Volume of a Sphere: Formula, Derivation & Solved Examples

What is Volume of Sphere?

Volume of a Sphere Formula

Derivation of the Volume of a Sphere

Arrhenius Method

Integration Method

How to Calculate Volume of a Sphere?

Solved Examples on Volume of Sphere

Things to Remember

Sample Questions

CBSE X Related Questions

1.In a right triangle ABC, right-angled at A, if $\sin B = \dfrac{1}{4}$, then the value of $\sec B$ is

2.Let $p$, $q$ and $r$ be three distinct prime numbers. Check whether $pqr + q$ is a composite number or not. Further, give an example for three distinct primes $p$, $q$, $r$ such that (i) $pqr + 1$ is a composite number (ii) $pqr + 1$ is a prime number

4.In the adjoining figure, $\triangle CAB$ is a right triangle, right angled at A and $AD \perp BC$. Prove that $\triangle ADB \sim \triangle CDA$. Further, if $BC = 10$ cm and $CD = 2$ cm, find the length of AD.

5.Prove that \(\dfrac{\sin \theta}{1 + \cos \theta} + \dfrac{1 + \cos \theta}{\sin \theta} = 2\csc \theta\)

6.From one face of a solid cube of side 14 cm, the largest possible cone is carved out. Find the volume and surface area of the remaining solid.Use $\pi = \dfrac{22}{7}, \sqrt{5} = 2.2$

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
In a right triangle ABC, right-angled at A, if $\sin B = \dfrac{1}{4}$, then the value of $\sec B$ is

2.
Let $p$, $q$ and $r$ be three distinct prime numbers. Check whether $pqr + q$ is a composite number or not. Further, give an example for three distinct primes $p$, $q$, $r$ such that
(i) $pqr + 1$ is a composite number
(ii) $pqr + 1$ is a prime number

4.
In the adjoining figure, $\triangle CAB$ is a right triangle, right angled at A and $AD \perp BC$. Prove that $\triangle ADB \sim \triangle CDA$. Further, if $BC = 10$ cm and $CD = 2$ cm, find the length of AD.

5.
Prove that \(\dfrac{\sin \theta}{1 + \cos \theta} + \dfrac{1 + \cos \theta}{\sin \theta} = 2\csc \theta\)

6.
From one face of a solid cube of side 14 cm, the largest possible cone is carved out. Find the volume and surface area of the remaining solid.
Use $\pi = \dfrac{22}{7}, \sqrt{5} = 2.2$