Volume of a Sphere

The volume of a sphere is the amount of space contained inside a sphere. It tells us how much material a sphere can hold. The volume of a sphere depends on its radius and is measured in cubic units such as cubic centimeters (cm³) or cubic meters (m³).

A solid sphere is completely filled from the inside. If the radius of the sphere is r, then its volume is given by:

Volume of a solid sphere = (4/3)πr³

where, r is the radius of the sphere and π is a constant whose value is taken as 22/7.

The volume of a sphere is the space contained inside it. A sphere can be of two types: solid sphere and hollow sphere.

Volume of a Hollow Sphere

A hollow sphere has an empty space inside it. It has two radii: the outer radius R and the inner radius r. The volume of a hollow sphere is found by subtracting the volume of the inner sphere from the volume of the outer sphere.

Volume of a hollow sphere = (4/3)π(R³ − r³)

Formula Derivation

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

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

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 = \int_{y=-R}^{y=R} dV

⇒ V = \int_{y=-R}^{y=R}π(R^2 - y^2)dy

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

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

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

⇒ V = \frac{4}{3}\pi R^3

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:

• Surface Area of Sphere
• Spherical Cap Volume Formula
• Spherical Sector Formula
• Spherical Segment Formula

Volume of Sphere Solved 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

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.