Grade 6
  1. Number System  1.1. Whole Numbers  1.1.1. Place Value and Face Value  1.1.2. Comparing and Ordering Numbers  1.1.3. Rounding Numbers  1.1.4. Properties of Whole Numbers (Associative, Commutative, Distributive)  1.2. Integers  1.2.1. Introduction to Integers  1.2.2. Addition and Subtraction of Integers  1.2.3. Multiplication and Division of Integers  1.2.4. Properties of Integer Operations  1.3. Fractions  1.3.1. Types of Fractions  1.3.2. Equivalent Fractions  1.3.3. Simplification of Fractions  1.3.4. Addition and Subtraction of Fractions  1.3.5. Multiplication and Division of Fractions  1.3.6. Comparing and Ordering Fractions  1.4. Decimals  1.4.1. Understanding Decimals  1.4.2. Conversion between Fractions and Decimals  1.4.3. Addition and Subtraction of Decimals  1.4.4. Multiplication and Division of Decimals  1.4.5. Comparing and Ordering Decimals
  2. Algebra  2.1. Basics of Algebra  2.1.1. Variables and Constants  2.1.2. Expressions and Equations  2.1.3. Algebraic Terms and Coefficients  2.2. Solving Simple Equations  2.2.1. Balancing Equations  2.2.2. Solving One-Step Equations  2.2.3. Solving Two-Step Equations  2.3. Patterns and Sequences  2.3.1. Identifying Patterns  2.3.2. Arithmetic Sequences
  3. Ratio and Proportion  3.1. Ratios  3.1.1. Understanding Ratios  3.1.2. Equivalent Ratios  3.1.3. Simplifying Ratios  3.2. Proportion  3.2.1. Introduction to Proportion  3.2.2. Direct and Inverse Proportions  3.3. Percentages  3.3.1. Converting Ratios to Percentages  3.3.2. Percentage Increase and Decrease  3.3.3. Discount and Profit Calculations  3.3.4. Simple Interest Calculations in Percentages in Ratio and Proportion
  4. Geometry  4.1. Basic Geometric Shapes  4.1.1. Points, Lines, and Angles  4.1.2. Types of Angles  4.1.3. Intersecting and Parallel Lines  4.2. Triangles  4.2.1. Types of Triangles  4.2.2. Properties of Triangles  4.2.3. Sum of Angles in a Triangle  4.2.4. Congruence of Triangles  4.3. Quadrilaterals  4.3.1. Types of Quadrilaterals  4.3.2. Properties of Quadrilaterals  4.4. Circles  4.4.1. Radius and Diameter  4.4.2. Circumference and Area  4.4.3. Arcs and Chords  4.4.4. Sector and Segment
  5. Mensuration  5.1. Perimeter  5.1.1. Perimeter of Simple Shapes  5.2. Area  5.2.1. Area of Rectangles and Squares  5.2.2. Area of Triangles and Parallelograms  5.2.3. Area of Composite Figures  5.2.4. Area of Trapezium  5.3. Volume  5.3.1. Volume of Cubes and Cuboids  5.3.2. Volume of Cylinders  5.3.3. Volume of Cones  5.3.4. Surface Area of Simple Solids
  6. Data Handling  6.1. Introduction to Data  6.1.1. Types of Data  6.1.2. Organizing Data  6.1.3. Frequency Distribution  6.2. Graphs and Charts  6.2.1. Bar Graphs  6.2.2. Line Graphs  6.2.3. Pictographs  6.2.4. Pie Charts  6.3. Mean, Median, and Mode  6.3.1. Calculating Mean  6.3.2. Finding Median  6.3.3. Identifying Mode  6.3.4. Range of Data in Mean, Median, and Mode
  7. Probability  7.1. Basics of Probability  7.1.1. Understanding Probability  7.1.2. Probability of Simple Events  7.1.3. Experimental and Theoretical Probability
  8. Practical Geometry  8.1. Construction of Shapes  8.1.1. Constructing Triangles  8.1.2. Constructing Quadrilaterals  8.1.3. Constructing Circles and Arcs  8.2. Symmetry  8.2.1. Line of Symmetry  8.2.2. Rotational Symmetry  8.2.3. Translational Symmetry

Grade 6 → Geometry → Circles ↓


Circumference and Area


What is a circle?

A circle is a perfectly round shape. It is the set of all points in a plane that are at a fixed distance from a given point called the center. The fixed distance from the center to any point on the circle is called the radius.

radius

Circumference of a circle

The circumference of a circle is the distance around it. Imagine that you measure the edge of the circle with a string and then measure the length of the string with a ruler; this length will be the circumference.

Perimeter formula

The formula to calculate the circumference (C) is:

C = 2 × π × r

where π (pi) is a special mathematical constant approximately equal to 3.14159, and r is the radius of the circle.

Example calculation

If the radius of a circle is 5 units, what is its circumference?

C = 2 × π × 5 ≈ 2 × 3.14159 × 5 ≈ 31.4159 units

Hence the circumference is approximately 31.42 units.

Understanding the diameter

The diameter is the second important part of a circle. It is twice the length of the radius and passes through the centre of the circle. Therefore, Diameter (d) = 2 × Radius (r)

Diameter

Formulas using diameter

Sometimes, you may know the diameter of a circle instead of its radius. In that case, the formula for circumference can be written as:

C = π × d

Example of calculation using diameter

If the diameter of a circle is 10 units, what is its circumference?

C = π × 10 ≈ 3.14159 × 10 ≈ 31.4159 units

Hence the circumference is again approximately 31.42 units.

Area of a circle

The area of a circle represents the space contained within its boundary. It's like figuring out how much paint you need to fill the entire circle.

Area formula

The formula to calculate area (A) is:

A = π × r²

Where r² means r squared, or r multiplied by itself.

Example calculation for area

If the radius of a circle is 7 units, what is its area?

A = π × 7² = π × 49 ≈ 3.14159 × 49 ≈ 153.938 units²

Hence the area is approximately 153.94 square units.

Why is π (pi) important?

π, pronounced "pi," is important for calculations involving circles. It is the ratio of any circle's circumference to its diameter and is the same no matter the size of the circle.

This ratio is approximately 3.14159, but it continues forever without repeating. Mathematicians often use 3.14 or 22/7 as an approximation of π.

A simple experiment

Take any circular object, such as a cup or bottle cap. Use a string to measure around the object. Then, measure its diameter (at the widest part) with a ruler. Now, divide the length of the circumference by the width:

Pi ≈ Circumference ÷ Diameter

You should get a result close to 3.14!

Bringing everyone together

Problem solving: Perimeter

Let's solve a problem: The radius of a wheel is 14 inches. How much distance does it travel in one complete revolution?

We need to find the circumference, because this is the distance it travels in one revolution.

C = 2 × π × r = 2 × π × 14 ≈ 2 × 3.14159 × 14 ≈ 87.9646 inches

Thus, the wheel moves approximately 87.96 inches.

Problem solving: Areas

Another problem: You have a circular garden with a radius of 10 meters. How much area will you need to plant grass?

Here we have to find the area of the circle.

A = π × r² = π × 10² = π × 100 ≈ 3.14159 × 100 ≈ 314.159 square meters

You will need enough grass for approximately 314.16 square meters.

Real-world applications

Understanding circles is important not only in math classes but also in everyday life. Engineers, architects, and even artists use the concepts of circumference and area.

Applications in engineering

Engineers use these formulas to design wheels, gears, and many circular components. Knowing the circumference helps understand how much distance a wheel will travel with each revolution, which is important for vehicles.

Applications in architecture

Architects use these concepts to design objects such as circular windows, domes and fountains, and ensure that they fit correctly into the design, because they know their area and the material needed around their perimeter.

Exercise exercises

Try these exercises to test your understanding:

  1. Find the circumference of a circle of radius 8 cm.
  2. What is the area of a circle with diameter 12 m?
  3. If the circumference of a circular table is 62.8 inches, what is its diameter?
  4. The radius of a circular park is 20 meters. Find its area in square meters.

Answer

Below are the solutions to the exercises:

  1. C = 2 × π × 8 ≈ 50.265 cm
  2. A = π × (12/2)² = π × 6² = π × 36 ≈ 113.097 m²
  3. C = π × d => d = C/π ≈ 62.8/3.14159 ≈ 20 inches
  4. A = π × 20² = π × 400 ≈ 1256.64 m²

These exercises help strengthen the understanding of finding perimeter and area.

Conclusion

Knowing how to work with the circumference and area of circles can help you solve real-world problems creatively and effectively. Whether a school project or a practical application, a clear understanding can make your work with circles much easier and more accurate.


Grade 6 → 4.4.2


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Radius and Diameter
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Circumference and Area

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