Volume of Sphere

The volume of a sphere tells us how much space is enclosed within a perfectly round 3D object. From sports balls to planets, spheres are everywhere, and calculating their volume is essential in both academic and real-world contexts. In this guide, you'll explore the volume of sphere formula, its step-by-step derivation, and real-life examples to help you understand and apply the concept easily.

 

Table of Contents  

• What is the Volume of Sphere?
• Volume of Sphere Formula with its Derivation
• Derivation of Volume of Sphere
• How to Calculate Volume of Sphere?
• Solved Examples on Volume of Sphere
• Practice Questions
• Conclusion
• FAQs On Volume of Sphere

 

What is the Volume of Sphere?  

The volume of a sphere measures the three-dimensional space inside it. A sphere is a perfectly symmetrical 3D shape where every point on the surface is the same distance from the center. The volume indicates how much space is contained within the sphere.  

To understand this better, think about filling a basketball or globe with water. The amount of water it can hold shows the volume of the sphere. Knowing the volume is important in fields like physics, engineering, and architecture. It also helps in practical tasks, like figuring out how much gas fits into a spherical balloon or how much fluid is in a spherical tank.

 

Volume of Sphere Formula with its Derivation  

The standard volume formula for a sphere is:  

Volume of Sphere = (4/3) × π × r³  

Where:  

• r = radius of  sphere 
• π ≈ 3.14159  

This formula calculates the sphere volume in cubic units based on its radius.

 

Derivation of Volume of Sphere  

Let’s go through the derivation of volume of a sphere using calculus.  

To derive the volume mathematically, imagine stacking very thin circular disks from the bottom to the top of the sphere. The radius of each disk changes based on its position.  

Consider a sphere with radius r, centered at the origin. The equation of the sphere in 3D is:  

x² + y² + z² = r²  

We revolve the semicircle y = √(r² - x²) around the x-axis to form a sphere. The volume generated by this rotation using integration is:  

V = π ∫ from -r to r of (r² - x²) dx  

= π [r²x - x³/3] from -r to r  

= π [2r³ - (2r³)/3] = (4/3) π r³  

Thus, the derivation of volume confirms that the volume of sphere is (4/3) π r³.

 

How to Calculate Volume of Sphere?  

To calculate the volume of a sphere,  you need either the radius of sphere or the diameter of sphere.  

Using the radius of sphere:  

If you know the radius, plug it into the formula:  

V = (4/3) π r³  

 

Using Diameter of Sphere:
If the diameter of sphere d is given:

r = d / 2  

V = (4/3) π (d/2)³ = (π d³) / 6  

 

Surface Area of Sphere  

While learning about the volume of a sphere, it’s also helpful to understand the sphere's surface area and total surface area of a sphere:  

Surface area of sphere = 4 π r²  

Though the surface area and volume are different, both are crucial for grasping the geometry of a sphere.

 

Solved Examples on Volume of Sphere  

Let’s look at a few examples to understand how to use the formula:  

Example 1:  

Find the volume of a sphere with a radius of 3 cm.  

V = (4/3) π (3)³
   = (4/3) π (27)
   = 36π ≈ 113.1 cm³  

 

Example 2:  

Find the volume of a sphere with a diameter of 10 cm.  

r = 10 / 2 = 5 cm  

V = (4/3) π (5)³
   = (4/3) π (125)
   = (500/3) π ≈ 523.6 cm³  

 

Example 3:  

A ball has a surface area of 314.16 cm². Find the volume of the sphere.  

First, find the radius:  

4π r² = 314.16 → r² = 314.16 / (4π) = 25 → r = 5  

V  = (4/3) π (5)³
    = (500/3) π ≈ 523.6 cm³  

These examples show how the formula works in different situations.

 

Practice Questions  

1. Try solving the following to master the volume of a sphere:  

2. Find the volume of a sphere with a radius of 7 cm.  

3. If the diameter of sphere is 12 inches, find the volume.  

4. A football has a total surface area of a spherearea of 452.16 cm². Calculate its volume.  

5. Derive the volume of the sphere using integration again.  

6. Solve and compare the surface area and volume for a radius of 6 cm.  

 

Conclusion  

Understanding the volume of a sphere is fundamental in geometry and has practical significance in many real-life applications, from engineering to everyday measurements. The standard volume formula, (4/3) × π × r³, helps us calculate the space a sphere occupies, whether we know the radius or the diameter.  

By exploring the derivation, learning how to apply the formula, reviewing examples, and solving practice problems, you will build a solid foundation on this topic. Additionally, knowing related concepts like surface area will deepen your understanding of spherical shapes.

 

Related Links

Volume of Cone - Learn how to calculate the volume of a cone using the standard formula, with visual illustrations and solved examples.

Volume of a Cylinder - Understand the concept and formula for finding the volume of a cylinder, with step-by-step examples to aid learning.

 

Frequently Asked Questions On Volume of Sphere  

1. What is CSA and TSA of a sphere?  

Ans: In the case of a sphere, the Curved Surface Area (CSA) and Total Surface Area (TSA) are the same since a sphere has no flat surfaces.

         The formula is:  

         CSA = TSA = 4πr²  

 

2. Why is 4:3 used in the volume of a sphere?  

Ans: The 4:3 ratio appears in the volume formula because of mathematical derivation involving integration and calculus. The volume formula:  

Volume = (4/3)πr³  

comes from revolving a semicircle around its diameter and summing up small volumes, resulting in the 4/3 coefficient.

 

3. What is 4 ⁄ 3 πr³?  

Ans: (4/3)πr³ is the formula for the volume of a sphere, where:  

• r = radius of  sphere  
• π ≈ 3.14159  

It represents the amount of three-dimensional space inside the sphere.

 

4. Why is a sphere 4 pi r²?  

Ans: 4πr² is the formula for the surface area of a sphere. It represents the total outer surface area covering the sphere. This value is derived using calculus by rotating a semicircle around its diameter and calculating the surface generated.

 

Understand and calculate math concepts like sphere volume with simple steps at Orchids The International School!