Grade 7
  1. Number System  1.1. Integers  1.1.1. Properties of Integers  1.1.2. Operations on Integers  1.1.3. Additive and Multiplicative Inverses  1.1.4. Applications of Integers  1.2. Rational Numbers  1.2.1. Definition of Rational Numbers  1.2.2. Properties of Rational Numbers  1.2.3. Operations on Rational Numbers  1.2.4. Standard Form of Rational Numbers  1.3. Decimals  1.3.1. Operations on Decimals  1.3.2. Converting Between Fractions and Decimals  1.4. Powers and Exponents  1.4.1. Laws of Exponents  1.4.2. Simplifying Expressions with Exponents  1.4.3. Scientific Notation  1.5. Roots  1.5.1. Square Roots  1.5.2. Cube Roots
  2. Algebra  2.1. Expressions  2.1.1. Algebraic Expressions  2.1.2. Simplifying Expressions  2.1.3. Evaluating Expressions  2.2. Linear Equations  2.2.1. Solving Simple Equations  2.2.2. Applications of Linear Equations  2.3. Algebraic Identities  2.3.1. Expanding Using Identities  2.3.2. Simplifying Using Identities
  3. Ratio and Proportion  3.1. Ratios  3.1.1. Understanding Ratios  3.1.2. Simplifying Ratios  3.1.3. Equivalent Ratios  3.2. Proportion  3.2.1. Concept of Proportion  3.2.2. Direct Proportion  3.2.3. Inverse Proportion  3.3. Unitary Method  3.3.1. Solving Problems Using the Unitary Method
  4. Geometry  4.1. Lines and Angles  4.1.1. Types of Angles  4.1.2. Pairs of Angles  4.1.3. Properties of Parallel Lines and a Transversal  4.1.4. Angle Bisectors  4.2. Triangles  4.2.1. Types of Triangles  4.2.2. Angle Sum Property  4.2.3. Exterior Angle Property  4.2.4. Pythagoras Theorem  4.3. Congruence of Triangles  4.3.1. Criteria for Congruence  4.3.2. Applications of Congruence  4.4. Quadrilaterals  4.4.1. Types of Quadrilaterals  4.4.2. Properties of Parallelograms  4.4.3. Special Quadrilaterals  4.4.4. Properties of Rectangles and Rhombuses
  5. Mensuration  5.1. Perimeter and Area  5.1.1. Perimeter of Plane Figures  5.1.2. Area of Plane Figures  5.1.3. Area of a Trapezium  5.1.4. Area of a Parallelogram  5.1.5. Area of a Circle  5.1.6. Area of Composite Figures  5.2. Surface Area and Volume  5.2.1. Cubes and Cuboids  5.2.2. Surface Area of Cylinders  5.2.3. Volume of Prisms and Cylinders  5.2.4. Volume and Surface Area of Cones
  6. Data Handling  6.1. Data Collection  6.1.1. Organizing Data  6.1.2. Frequency Distribution  6.2. Graphical Representation of Data  6.2.1. Bar Graphs  6.2.2. Double Bar Graphs  6.2.3. Pie Charts  6.2.4. Histograms  6.2.5. Line Graphs  6.3. Probability  6.3.1. Simple Events  6.3.2. Experimental Probability  6.3.3. Theoretical Probability
  7. Practical Geometry  7.1. Construction of Triangles  7.1.1. Constructing Triangles Given Side Lengths  7.1.2. Constructing Triangles Given Side and Angle  7.1.3. Constructing Right-Angled Triangles  7.2. Construction of Quadrilaterals  7.2.1. Quadrilaterals Given Side Lengths and Angles  7.2.2. Constructing Parallelograms and Rectangles

Grade 7 → Mensuration → Surface Area and Volume ↓


Surface Area of Cylinders


When we study the world around us in terms of shapes and measurements, we come across many objects that have a cylindrical shape. Cylindrical shapes are very common in both nature and man-made objects. Examples include trees, cans, pipes, and even roller coasters. Today, we will explore how to find the surface area of these cylinders, so that it is easier to understand.

What is a cylinder?

A cylinder is a three-dimensional figure that has two parallel circles of equal size at the top and bottom. These circles are called the bases of the cylinder. The part connecting the two bases is round and is called the lateral surface. If you imagine a soup can, consider the top and base as two circular faces and the curved label around the can as the lateral surface.

                ,
               ,
              ,
             ,
            ,
           ,

Components of the cylinder

1. Aadhaar

The cylinder has two bases, which are circles. These circular bases are perpendicular and parallel to each other. The distance between the two bases is called the height of the cylinder.

2. Height

This is the distance between the two circular bases. It is also the length of the perpendicular line segment connecting the centers of the circles.

3. Radius

The radius of the cylinder is the radius of the circular base. It is the distance from the center of the base to the edge.

4. Lateral surface

This is the curved surface connecting the two bases. If you cut the lateral surface from top to bottom and flatten it, you will get a rectangle.

          ,      
         /base |      
        | | Height 
         ,       
              Base

Visualizing the surface area of a cylinder

The surface area of a cylinder is the total area of all its faces. In simple words, it is the space occupied by the outer surface of the cylinder.

If you open up the edge of a cylinder and flatten it, this is what you will see:

    ,
    | side | <-- it's a rectangle
    ,
      ugh
      ,

It is calculated as follows:

Total Surface Area Formula

The total surface area of a cylinder is the sum of the areas of its two circular bases and the lateral surface area. Below is the formula used to calculate it:

 Total Surface Area = 2πr² + 2πrh

Where:

  • r is the radius of the base
  • h is the height of the cylinder
  • π is a constant approximately equal to 3.14159

Understanding each part of the formula

2πr² calculates the area of both circular bases. The formula for the area of a circle is πr². Since there are two bases, you multiply by 2. The part 2πrh stands for the lateral surface area. This can be thought of as the side opening (which is shaped like a rectangle) where the length is the perimeter of the base (2πr) and the width is the height (h).

Step-by-step example

Let's solve this together!

Consider a cylinder with a radius of 3 cm and a height of 5 cm. Let's use the formula to find its surface area.

  1. Calculate the area of the circular bases:
     2πr² = 2 * π * 3² = 2 * 3.14159 * 9 = 56.54867 cm²
  2. Calculate the lateral surface area:
     2πrh = 2 * π * 3 * 5 = 2 * 3.14159 * 15 = 94.2477 cm²
  3. Add the two results to find the total surface area:
     Total Surface Area = 56.54867 + 94.2477 = 150.79637 cm²

Another example

Suppose a cylindrical water tank has a radius of 7 m and a height of 10 m. Calculate the surface area.

  1. Calculate the area of the circular bases:
     2πr² = 2 * π * 7² = 2 * 3.14159 * 49 = 307.87698 m²
  2. Calculate the lateral surface area:
     2πrh = 2 * π * 7 * 10 = 2 * 3.14159 * 70 = 439.82297 m²
  3. Add the two results to find the total surface area:
     Total Surface Area = 307.87698 + 439.82297 = 747.69995 m²

Break it down further

Let's look in further detail at how these components work together in a visual format. Imagine that you cut the side surface of the cylinder along its height and roll it out flat; you get a rectangle that has:

  • Length = circumference of the base, i.e. 2πr
  • Width = Height of the cylinder = h
    ,
    |side | <- Rectangle: Length = 2πr, Width = h
    ,
      oo <- circle : area = πr² each
      ,

For a simple exercise, think of wrapping a label around a can. The length of the label covers the circumference of the can, and the height of the label is equal to the height of the can.

Practical applications

Why do we learn about the surface area of a cylinder? Knowing how to calculate surface area can be incredibly useful in real life. Some applications include:

  • To determine the amount of material required to make a cylindrical container.
  • Painting or coating jobs where you need to know exactly how much area you will be covering.
  • Packaging design, to estimate how much packaging material is needed.

Practice problems

  1. Find the total surface area of a cylinder of radius 4 cm and height 6 cm.
  2. A soup can has a height of 10 cm and a radius of 3.5 cm. What is its surface area?
  3. The radius of a chemical drum is 1.5 m and height is 2 m. Find its surface area.

Conclusion

The surface area of a cylinder is easy to calculate once we break it down into its components - the two circular bases and the side rectangle. Understanding and calculating surface area gives invaluable information when dealing with any cylindrical shape in real life, from gift wrapping to industrial applications.

Whether you're a student, a painter, or someone who works in logistics, the ability to understand and calculate surface area is a useful skill in the practical world.

Additional tips for students

When solving problems and working with the surface area of cylinders, remember:

  • Make sure all your measurements are in the same units.
  • Keep your math neat and organized to avoid mistakes.
  • Practice with different types of cylinders with different radii and heights.
  • Consider using real-world objects to enhance understanding.

With practice, calculating the surface area of cylinders will become a simple task. Keep practicing and exploring, and remember to use these math skills to observe and measure the world around you!

Answers to exercise problems

  1. Total surface area = 2π(4)² + 2π(4)(6) = 100.53096 + 150.796 = 251.327 sq. cm
  2. Total surface area = 2π(3.5)² + 2π(3.5)(10) = 76.96902 + 219.91149 = 296.8805 sq. cm
  3. Total surface area = 2π(1.5)² + 2π(1.5)(2) = 14.13717 + 18.84956 = 32.98673 m²

Grade 7 → 5.2.2


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Cubes and Cuboids
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Volume of Prisms and Cylinders

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Surface Area of Cylinders

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