Grade 8
  1. Number Systems  1.1. Rational Numbers  1.2. Irrational Numbers  1.3. Real Numbers  1.4. Properties of Real Numbers  1.4.1. Commutative Property  1.4.2. Associative Property  1.4.3. Distributive Property  1.5. Representation on the Number Line  1.6. Operations on Real Numbers  1.6.1. Addition and Subtraction  1.6.2. Multiplication and Division  1.7. Rationalization  1.8. Exponents and Powers  1.9. Square Roots and Cube Roots
  2. Algebra  2.1. Algebraic Expressions  2.2. Identities and Simplification  2.2.1. Multiplying Binomials  2.2.2. Using Algebraic Identities  2.3. Factorization  2.3.1. Common Factors  2.3.2. Factorization by Grouping  2.3.3. Factorization of Trinomials  2.3.4. Special Products  2.4. Linear Equations in One Variable  2.4.1. Solving Linear Equations  2.4.2. Word Problems on Linear Equations
  3. Geometry  3.1. Understanding Quadrilaterals  3.1.1. Properties of Quadrilaterals  3.1.2. Types of Quadrilaterals  3.1.2.1. Parallelogram  3.1.2.2. Rectangle  3.1.2.3. Square  3.1.2.4. Rhombus  3.1.2.5. Trapezium  3.2. Constructions  3.2.1. Constructing Quadrilaterals  3.2.2. Constructing Bisectors  3.2.3. Constructing Angles  3.2.4. Constructing Triangles  3.3. Symmetry and Transformations  3.3.1. Reflection  3.3.2. Rotation  3.3.3. Translation  3.3.4. Symmetry in Patterns
  4. Mensuration  4.1. Area and Perimeter  4.1.1. Perimeter of Polygons  4.1.2. Area of Triangles  4.1.3. Area of Quadrilaterals  4.1.4. Area of Circles  4.2. Surface Area and Volume  4.2.1. Cubes  4.2.2. Cuboids  4.2.3. Cylinders  4.2.4. Cones  4.2.5. Spheres
  5. Data Handling  5.1. Organizing Data  5.2. Frequency Distribution Tables  5.3. Graphical Representation of Data  5.3.1. Bar Graphs  5.3.2. Histograms  5.3.3. Pie Charts  5.3.4. Line Graphs (added)  5.4. Probability  5.4.1. Introduction to Probability  5.4.2. Experimental Probability
  6. Coordinate Geometry  6.1. Cartesian Plane  6.2. Plotting Points  6.3. Coordinate System  6.4. Distance Between Points  6.5. Applications of Coordinate Geometry
  7. Introduction to Graphs  7.1. Linear Graphs  7.2. Applications of Graphs  7.3. Distance-Time Graphs  7.4. Speed-Time Graphs
  8. Comparing Quantities  8.1. Ratio and Proportion  8.2. Percentage  8.2.1. Increase and Decrease Percent  8.2.2. Discount in Percentage  8.2.3. Profit and Loss in Percentage  8.2.4. Simple Interest  8.2.5. Compound Interest
  9. Exponents and Powers  9.1. Laws of Exponents  9.2. Negative Exponents  9.3. Standard Form  9.4. Applications of Exponents
  10. Introduction to Squares and Square Roots  10.1. Properties of Square Numbers  10.2. Finding Square Roots  10.2.1. Prime Factorization Method  10.2.2. Division Method  10.3. Estimating Square Roots
  11. Introduction to Cubes and Cube Roots  11.1. Properties of Cube Numbers  11.2. Finding Cube Roots  11.2.1. Prime Factorization Method in Finding Cube Roots

Grade 8 → Mensuration → Surface Area and Volume ↓


Spheres


Today, we will explore one of the most fascinating shapes in geometry: the sphere. Have you ever wondered about the shape of a basketball, a soap bubble, or the planet we live on? These everyday objects look like spheres. Let's take a deeper look at what spheres are, how to find their surface area, and how to find their volume.

What is sphere?

A sphere is a perfectly round three-dimensional shape. Every point on the surface of a sphere is at the same distance from its center. This constant distance is called the radius of the sphere.

radius Hey

In the figure above, the blue shape is a sphere. The red line represents the radius, denoted by r, and 'O' is the center of the sphere.

Surface area of a sphere

The surface area of a sphere is the total area covered by the outer surface of the sphere. This can be visualized as the total area you would need to cover the sphere if it were a physical model.

Surface area formula

The formula for finding the surface area of a sphere is:

Surface area = 4πr2

Here, π (pi) is approximately equal to 3.14159, and r is the radius of the sphere.

Example calculation

Example 1: Let us find the surface area of a sphere with radius 7 cm.

Surface area = 4πr2
            = 4 × 3.14159 × (7)2
            = 4 × 3.14159 × 49
            = 4 x 153.93804
            = 615.75216 cm2

Therefore, the surface area of the sphere is approximately 615.75 cm2.

Volume of a sphere

The volume of a sphere refers to the amount of space present inside it. Imagine filling the sphere with a liquid; measuring that liquid gives us an idea of its volume.

Volume formula

The formula for finding the volume of a sphere is:

Volume = (4/3)πr3

Again, π is approximately 3.14159, and r is the radius of the sphere.

Example calculation

Example 2: Let us find the volume of a sphere with radius 6 cm.

Volume = (4/3)πr3
       = (4/3) × 3.14159 × (6)3
       = (4/3) × 3.14159 × 216
       = 4 × 3.14159 × 72
       = 904.77868 cm3

Therefore, the volume of the sphere is approximately 904.78 cm3.

Visualization of area calculations

Let us understand these formulas with another example. Consider a sphere of radius 5 units. Using the formulas we have learned, we can find its surface area and volume.

5 Hey

Surface area calculation:

Surface area = 4πr2
            = 4 × 3.14159 × (5)2
            = 4 × 3.14159 × 25
            = 314.159 cm2

The surface area is approximately 314.16 cm2.

Volume calculation:

Volume = (4/3)πr3
       = (4/3) × 3.14159 × (5)3
       = (4/3) × 3.14159 × 125
       = 523.598 cm3

The volume is approximately 523.60 cm3.

Why are surface area and volume important?

Understanding the surface area and volume of a sphere is important in a variety of real-life applications. For example, architects and engineers consider surface area when designing objects such as domes or storage tanks to optimize material use. Similarly, knowing the volume helps determine how much space is inside, which is important for packaging or manufacturing processes.

Real life applications of shells

There are many examples of spheres in our daily life. Here are some examples:

  • Planets and moons: The Earth, Moon and other planets are almost perfect spheres.
  • Sports balls: Basketballs, footballs, and other sports balls are spherical.
  • Soap bubbles: A thin film of soap takes the shape of a sphere due to surface tension.
  • Ornaments: The most commonly used round ornaments for decoration are shells.

Conclusion

Spheres are fascinating shapes whose properties and applications make them essential in both mathematics and a variety of industries. By understanding how to calculate their surface area and volume, you can appreciate their presence and importance in the world around you.

I am confident that this comprehensive guide to the fields, including concepts, examples, and practical applications, will improve your understanding of this mathematical subject. Keep exploring the world of mathematics to uncover even more wonders!


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