Formula for Surface Area and Volume of a Sphere with Examples
The concept of sphere plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. You'll often come across spheres in geometry, physics, and daily objects like balls, bubbles, and planets. Understanding the sphere’s properties and formulas is essential for board exams and general problem-solving.
What Is Sphere?
A sphere is a perfectly round three-dimensional object where every point on its surface is equidistant from its center. Unlike a circle (which is a flat 2D shape), a sphere extends in all directions, making it a 3D solid. You’ll find this concept applied in area and volume calculation, geometric constructions, and various competitive exams.
Key Formula for Sphere
Here’s the standard formula for the volume and surface area of a sphere, where \( r \) is the radius:
| Property | Formula |
|---|---|
| Volume (V) | \( V = \frac{4}{3}\pi r^3 \) |
| Surface Area (A) | \( A = 4\pi r^2 \) |
Properties of Sphere
- All surface points are equidistant from the center (radius).
- Has no edges or vertices—only one curved surface.
- Perfectly symmetrical and not a polyhedron.
- Same diameter at all cross-sections.
- Volume and area depend only on the radius.
Cross-Disciplinary Usage
Sphere is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. For example, in Physics, the sphere formula is used to calculate the volume of bubbles or planets. Students preparing for JEE or NEET will see its relevance in various questions.
Solved Example: Finding Volume
Let’s see how to use the formula to solve a board-level question.
Question: Find the volume of a sphere with a radius of 7 cm. (Use \( \pi = 22/7 \))
2. Formula: \( V = \frac{4}{3} \pi r^3 \)
3. Substitute values: \( V = \frac{4}{3} \times \frac{22}{7} \times 7^3 \)
4. \( 7^3 = 343 \). Now: \( V = \frac{4}{3} \times \frac{22}{7} \times 343 \)
5. \( \frac{22}{7} \times 343 = 22 \times 49 = 1078 \)
6. \( V = \frac{4}{3} \times 1078 = \frac{4312}{3} \approx 1437.33 \) cm³
Final Answer: The volume of the sphere is 1437.33 cm³.
Types of Sphere
- Solid Sphere: Completely filled 3D object (e.g., shot put ball).
- Hollow Sphere: Only the surface is present, like a football or soap bubble.
Difference Between Sphere and Circle
| Sphere | Circle |
|---|---|
| 3D shape with all points equidistant from the center | 2D flat shape with points at a fixed radius from the center |
| Has volume and surface area | Has area and perimeter (circumference) |
| Example: globe, ball | Example: coin, plate |
Real-Life Examples of Spheres
- Football and tennis ball
- Water droplets
- Planets like Earth
- Bubbles
- Marbles
Speed Trick or Vedic Shortcut
If the radius of a sphere doubles, its volume increases by 8 times! Since volume is proportional to the cube of the radius, remember: \( (2r)^3 = 8r^3 \). Use this trick for quick reasoning questions in your Olympiad or competitive exams. Vedantu’s live classes regularly teach such easy shortcuts.
Practice Problems: Try These Yourself
- Find the surface area of a sphere with radius 5 cm. (\( \pi = 3.14 \))
- If a sphere has a volume of 904.32 cm³, what is its radius? (\( \pi = 3.14 \))
- Give 3 real-life examples of a hollow sphere.
- What is the difference between the surface area of a sphere and that of a hemisphere of radius r?
Frequent Errors and Misunderstandings
- Using the circle’s area formula instead of the sphere’s surface area.
- Confusing diameter and radius in formula application.
- Missing units (cm² for area, cm³ for volume).
- Forgetting the difference between 2D and 3D shapes.
Relation to Other Concepts
The idea of sphere connects closely with topics such as 3D Shapes and Surface Area and Volume. Mastering spheres makes it easier to solve questions on cylinders, cones, and hemispheres in later chapters. For more on how spheres are used with other solids, check this guide on solids combinations.
Classroom Tip
A quick way to remember sphere formulas: “Four Pi R Square for surface area, and Four-Thirds Pi R Cube for volume.” Say it aloud a few times! Vedantu’s teachers use such memory aids in their online live classes to make maths fun and easy to retain.
We explored sphere—from definition, formula, examples, mistakes, and connections to other subjects. Continue practicing with Vedantu to become confident in solving surface area and volume questions about spheres and other 3D shapes. For instant revision, check out the Class 10 Maths Formula Sheet including all sphere formulas.
FAQs on Sphere in Maths Definition Formula and Properties
1. What is a sphere in Maths?
A sphere is a three-dimensional shape consisting of all points in space that are at a fixed distance from a fixed point called the centre.
- The fixed distance is called the radius.
- A sphere is perfectly round in 3D, like a ball.
- It has no edges, no vertices, and no flat faces.
2. What is the formula for the volume of a sphere?
The volume of a sphere is given by the formula V = (4/3)πr³.
- r is the radius of the sphere.
- π ≈ 3.1416.
V = (4/3)π(3)³ = (4/3)π(27) = 36π ≈ 113.1 cm³.
3. What is the surface area of a sphere?
The surface area of a sphere is calculated using the formula A = 4πr².
- r is the radius.
- This gives the total curved surface area.
A = 4π(5)² = 4π(25) = 100π ≈ 314.16 cm².
4. What is the difference between a sphere and a circle?
The main difference is that a circle is a 2D shape, while a sphere is a 3D shape.
- A circle lies on a flat plane and has area but no volume.
- A sphere occupies space and has volume and surface area.
- A circle has circumference; a sphere does not.
5. How do you find the radius of a sphere from its volume?
To find the radius from volume, use the rearranged formula r = ∛(3V / 4π).
- Start with V = (4/3)πr³.
- Rearrange to r³ = 3V / 4π.
- Take the cube root.
r³ = (3 × 288π) / 4π = 216,
r = ∛216 = 6 cm.
6. What is the equation of a sphere in coordinate geometry?
The standard equation of a sphere with centre (h, k, l) and radius r is (x − h)² + (y − k)² + (z − l)² = r².
- (h, k, l) represents the centre coordinates.
- r is the radius.
7. Why is the surface area of a sphere 4πr²?
The surface area of a sphere is 4πr² because it is mathematically proven using calculus and geometric principles.
- It comes from integrating circular cross-sections.
- Interestingly, it equals four times the area of a circle with the same radius (πr²).
8. How many faces, edges, and vertices does a sphere have?
A sphere has 0 faces, 0 edges, and 0 vertices.
- It has only one continuous curved surface.
- There are no flat surfaces (faces).
- There are no corners (vertices) or line segments (edges).
9. What is a hemisphere?
A hemisphere is half of a sphere formed by cutting it with a plane through its centre.
- Volume = (2/3)πr³.
- Curved surface area = 2πr².
- Total surface area (including base) = 3πr².
10. What are some real-life examples of a sphere?
Common real-life examples of a sphere include balls, globes, and bubbles.
- A football or basketball approximates a sphere.
- The Earth is nearly spherical in shape.
- Soap bubbles form spheres due to surface tension.
