Description:
The chapter “Sphere / গোলক” in the Class 10 Mathematics syllabus under the West Bengal Board introduces students to one of the most symmetrical 3D shapes in geometry — the sphere. In this chapter, students learn about the properties of a sphere, its key components like radius and diameter, and how to calculate surface area and volume. The chapter also distinguishes spheres from other solids like cylinders and cones through practical mathematical examples.
1. Introduction to Sphere
A sphere (গোলক) is a three-dimensional solid where all points on its surface are equidistant from a fixed central point. This distance is known as the radius. The most common real-life example of a sphere is a ball. Unlike other 3D shapes, a sphere has no flat surfaces, edges, or vertices.
Mathematical Definition:
A sphere is the set of all points in space that are at a constant distance (radius) from a fixed point (centre).
2. Important Terms and Components
2.1 Centre (কেন্দ্র):
The fixed point inside the sphere from which all surface points are equidistant. It acts as the origin for calculating radius, surface area, and volume.
2.2 Radius (ব্যাসার্ধ):
The line segment joining the centre to any point on the surface of the sphere.
Notation: Usually denoted as r.
Example: If the radius of a sphere is 7 cm, every point on the surface is exactly 7 cm from the centre.
2.3 Diameter (ব্যাস):
A straight line passing through the centre and touching two points on the surface.
Formula: Diameter (d) = 2 × radius (r)
Example: If r = 5 cm, then d = 2 × 5 = 10 cm.
2.4 Surface Area (পৃষ্ঠতল ক্ষেত্রফল):
The total outer area of the spherical surface.
Formula:
Surface Area = 4πr²
Example: For r = 3 cm, Surface Area = 4 × π × 3² = 113.04 cm² (approx).
2.5 Volume (আয়তন):
The space enclosed within the sphere.
Formula:
Volume = (4/3)πr³
Example: For r = 3 cm, Volume = (4/3) × π × 3³ = 113.1 cm³ (approx).
3. Properties of a Sphere
3.1 Perfect Symmetry:
A sphere is perfectly symmetrical around its centre. Every axis passing through the centre divides the sphere into two equal halves.
3.2 No Edges or Vertices:
Unlike a cube or pyramid, a sphere does not have edges, faces, or corners. This property makes it unique among 3D solids.
3.3 Equal Cross-Sections:
Any cross-section through the centre of a sphere forms a perfect circle. This is useful in applications like tomography or design modelling.
4. Surface Area of a Sphere – In-Depth Understanding
The surface area of a sphere represents the amount of two-dimensional space covering its outer shell.
Derivation Insight (Syllabus-Based):
Though the derivation is not required at Class 10 level, understanding the formula and application is essential.
If r = radius of the sphere,
then Surface Area = 4πr².
Practical Example:
Let the radius of a football be 10 cm.
Then Surface Area = 4 × 3.1416 × (10)² = 1,256.64 cm²
5. Volume of a Sphere – Concept and Calculation
Volume indicates how much space is enclosed inside the sphere. The formula is slightly more complex but crucial in practical scenarios like fluid displacement or volume storage.
Formula:
Volume = (4/3)πr³
Example Calculation:
Let the radius of a rubber ball be 7 cm.
Volume = (4/3) × π × 7³ = (4/3) × 3.1416 × 343 ≈ 1436.76 cm³
6. Units and Conversions
6.1 Surface Area Units:
Always expressed in square units (e.g., cm², m²)
6.2 Volume Units:
Always expressed in cubic units (e.g., cm³, m³)
6.3 Conversion Tips:
- 1 m = 100 cm
- 1 m² = 10,000 cm²
- 1 m³ = 1,000,000 cm³
7. Comparison with Other Solids
| Shape | Surface Area Formula | Volume Formula | Flat Faces |
| Sphere | 4πr² | (4/3)πr³ | No |
| cylinder | 2πr(h + r) | πr²h | Yes |
| Cone | πr(l + r) | (1/3)πr²h | Yes |
8. Practical Applications of Spheres
- Astronomy: Planets and stars are roughly spherical due to gravitational pull.
- Sports: Balls used in games like cricket, football, and tennis are spherical.
- Engineering: Spherical shapes are used in pressure vessels and mechanical components.
- Architecture: Domes and globes often reflect spherical geometry.
9. Summary
The concept of a sphere introduces students to symmetrical, smooth 3D shapes that are common in both natural and human-made structures. With clearly defined formulas for surface area (4πr²) and volume ((4/3)πr³), this chapter equips students with tools to analyze and calculate properties of spheres in mathematical problems.
Mastering these concepts also lays the groundwork for advanced geometry in higher classes and real-world problem-solving.