Surface Area and Volume Formula of a Sphere with Solved Example
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: Meaning, Properties, and Formulas in Maths
1. What is a sphere in Maths?
In mathematics, a sphere is a perfectly round three-dimensional geometric object. Every point on its surface is equidistant from a single central point. Think of a ball; that's a real-world example of a sphere.
2. What is the formula for the volume and surface area of a sphere?
The formulas are: Volume (V) = (4/3)πr³ and Surface Area (A) = 4πr², where r represents the radius of the sphere (the distance from the center to any point on the surface).
3. What is the difference between a sphere and a circle?
A circle is a two-dimensional shape; it's a flat, round figure. A sphere is three-dimensional; it's a round solid object. Think of a circle as a cross-section of a sphere.
4. How do you solve word problems involving spheres?
First, identify the given information (usually radius, diameter, volume, or surface area). Then, select the appropriate formula (volume or surface area) and substitute the known values. Solve the equation to find the unknown quantity. Remember to use the correct units.
5. Give 5 examples of spheres seen in daily life.
Examples include: a basketball, a globe, a marble, a ball bearing, and a raindrop (approximately).
6. What are the properties of a sphere?
Key properties include: It's perfectly symmetrical, all points on the surface are equidistant from the center (radius), it has a constant width and circumference, and it possesses a single center point.
7. How is the surface area formula of a sphere derived?
The derivation typically involves calculus (integration) by considering infinitesimal surface elements and summing them over the entire sphere. A simplified explanation often uses the relationship between the sphere’s surface area and the circumscribed cylinder.
8. What is the effect of doubling the radius on the sphere’s volume?
Doubling the radius increases the volume by a factor of eight (2³ = 8). This is because volume is proportional to the cube of the radius.
9. What is a hemisphere?
A hemisphere is exactly half of a sphere. It is created by slicing a sphere in half through its center.
10. How do I calculate the diameter of a sphere given its volume?
First, use the volume formula (V = (4/3)πr³) to solve for the radius (r). Then, double the radius to find the diameter (diameter = 2r).
11. What is the relationship between a sphere and a great circle?
A great circle is the largest possible circle that can be drawn on the surface of a sphere. It always passes through the center of the sphere. Any plane passing through the center of a sphere will intersect the sphere's surface along a great circle.
12. What are some real-world applications of sphere calculations?
Sphere calculations are used in various fields, including: calculating the volume of spherical tanks, determining the surface area of planets, designing ball bearings, and modeling astronomical objects.
