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 ↓


Cylinders


In the study of measurement, a branch of mathematics concerned with measurement, a cylindrical figure is a three-dimensional figure with a curved surface and two parallel circular bases. Understanding the surface area and volume of a cylinder is important for calculating and estimating the space required for countless real-world applications, such as designing objects such as tanks, jars, and tubes.

Understanding the cylinder

First, let's define a cylinder a little more formally. A cylinder has two distinct parts: the circular base and the curved surface. The bases are identical circles, meaning they have the same size and shape. The height of the cylinder is the distance between these two circular bases.

It is also important to note the radius of the cylinder, which we denote by r. The radius is the distance from the center of a base to its edge. The height is usually denoted by h. With these elements we can calculate both the surface area and the volume of the cylinder.

Surface area of a cylinder

The surface area of a cylinder is made up of three pieces: the area of the two circular bases and the area of the curved surface. Let's find the total surface area step by step.

Area of circular bases

The area of a circle is calculated using the following formula:

    Area of a circle = πr²

Since a cylinder consists of two such circles, the total area from its two bases is given by:

    Area of two bases = 2 × πr²

Area of curved surface

The curved surface, when "unfolded" or "opened out", forms a rectangle. The length of this rectangle is equal to the circumference of the base (circle), which is 2πr, and the width is the height h of the cylinder.

Circumference: 2πr Height: h

Therefore, the area of the curved surface is:

    Curved surface area = 2πr × h

Total surface area of a cylinder

By adding the areas of the bases and the curved surface, we can calculate the total surface area of a cylinder:

    Total surface area = 2πr² + 2πrh
                       = 2πr(r + h)

Volume of a cylinder

The volume of an object tells how much space is inside its surface. For a cylinder, the volume is the space inside the object and is determined by the base area and the height.

To find the volume, we multiply the area of the base by the height. The area of the base is the area of a circle:

    Area of the base = πr²

Multiplying by the height h gives the formula for the volume of a cylinder:

    Volume of cylinder = πr²h

Example

Let us use some practical examples to understand the use of these formulas better.

Example 1: Finding the surface area

Suppose we have a cylinder with a radius of 3 cm and a height of 5 cm. To find its surface area, we use the formula:

    Surface area = 2πr(r + h)

Enter numbers:

    Surface area = 2π × 3 × (3 + 5)
                 = 2π × 3 × 8
                 = 48π cm²

Simplifying in terms of π, or using the approximation π ≈ 3.14, gives approximately:

    Surface area ≈ 48 × 3.14 ≈ 150.72 sq.cm

Example 2: Calculating volume

Consider the same cylinder with a radius of 3 cm and a height of 5 cm. To find its volume, we use the formula:

    Volume = πr²h

Enter values:

    Volume = π × 3² × 5
           = π × 9 × 5
           = 45π cm³

Approximation using π:

    Volume ≈ 45 × 3.14 ≈ 141.3 cm³

Volume comparison example

Consider two cylinders: cylinder A with a radius of 4 cm and a height of 10 cm, and cylinder B with a radius of 5 cm and a height of 8 cm. Let's calculate and compare their volumes.

First, cylinder A:

    Volume of cylinder A = π × 4² × 10
                         = π × 16 × 10
                         = 160π cm³
    Approximating using π ≈ 3.14,
                         ≈ 160 × 3.14 ≈ 502.4 cm³

Now, cylinder B:

    Volume of cylinder B = π × 5² × 8
                         = π × 25 × 8
                         = 200π cm³
    Approximating using π ≈ 3.14,
                         ≈ 200 × 3.14 ≈ 628 cm³

Conclusion: The volume of cylinder B is larger than that of cylinder A.

Understanding cylinders, their surface areas, and their volumes provides a foundational skill for measuring and constructing objects in the real world. Whether you're designing practical storage solutions or building complex engineering structures, these formulas are essential tools.


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Cylinders

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