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 ↓


Sector and Segment


Circles are fascinating shapes that we come across in everyday life. Be it a round clock, a pizza or the wheels of a car, circles are all around us. In this explanation, we will learn about two important characteristics of a circle: sector and segment. Let us break down these concepts and understand them in detail.

Circle basics

Before moving on to sectors and segments, it's important to know some basic parts of a circle. A circle is a shape whose all points are the same distance from its center. This distance is called the radius. A line that goes all the way around the circle, passing through its center, is called the diameter. The circumference is the total distance around the circle.

What is a sector?

A sector is like a "slice" of a circle. Imagine a pizza. When you cut a slice of pizza, that slice is a sector of the pizza circle. A sector is formed between two radii and the connecting arc.

For a regular circle, try dividing it into several slices, like a pie chart. Each slice, or "sector", is defined by two radii and the arc between them. The shape of a sector is often described by the angle made by the radii at the center of the circle, known as the central angle.

Area

arc is the portion of the circle's circumference that connects the two radii of the sector. radii are the two straight lines from the center that form part of the sector boundary.

Calculating the area of a sector

The area of a sector can be calculated if we know the radius and central angle of the sector. The formula for the area of a sector is:

Area of Sector = (Central Angle / 360) * π * radius²

Where:

  • The central angle is the angle in degrees between the two radii.
  • π (pi) is approximately 3.14159.
  • The radius is the distance from the center to the boundary of the circle.

For example, if the central angle of a sector is 60 degrees and the radius is 10 units, the area of the sector would be calculated as follows:

Area of Sector = (60 / 360) * π * 10² = (1/6) * π * 100 = 16.67π

What is a segment?

A segment in a circle is like a "cap" or "a piece with a curved edge". It is the area between a chord and the arc it forms.

To visualize a segment, think of a circle and a line inside the circle that does not pass through the center. This line is called a chord. The segment is the area between this line (the chord) and the arc of the circle above it.

Section

chord is a straight line connecting two points on the boundary of a circle, and arc is the curved portion of the circumference between those two points.

Calculating the area of a segment

Finding the area of a segment is a little more complicated than finding a sector. Usually, it involves subtracting the area of the triangular portion beneath the arc from the area of the sector that includes both the arc and the chord.

Area of Segment = Area of Sector - Area of Triangle

This involves using trigonometric functions and often understanding geometry beyond the basics, but the important thing to know is that the segment is simply a part of the sector with the triangle removed.

Text examples and practice questions

Example 1: Understanding regions

Suppose we have a circle with a radius of 7 units, and we want to find the area of a sector with a central angle of 45 degrees. Follow the formula:

Area of Sector = (45 / 360) * π * 7² = (1/8) * π * 49 = 6.125π

Therefore, when π is approximately 3.14159, the area of the sector is about 19.24 square units.

Example 2: Understanding clauses

Consider a circle with radius 10 units. Let's find the area of a sector with a central angle of 90 degrees.

First, find the area of a sector with a 90 degree central angle:

Area of Sector = (90 / 360) * π * 10² = (1/4) * π * 100 = 25π

Then, find the area of the triangular part of the segment:

Area of Triangle = (1/2) * radius * radius * sin(Central Angle) = (1/2) * 10 * 10 * sin(90 degrees) = 50 (since sin(90) = 1)

Finally, subtract the area of the triangle from the area of the sector:

Area of Segment = 25π - 50

This example gives us some insight into how to calculate these areas in practical situations.

Conclusion

Sectors and segments are two interesting and important parts of a circle. Sectors, like slices of pizza, are the regions formed between two radii and an arc, while segments are the regions bounded by a chord and its corresponding arc. Understanding how to measure and calculate the areas of these parts requires knowledge of central angles, basic geometry, and trigonometry.

By exploring and practicing problems involving areas and volumes, you will become more familiar and comfortable with these concepts, building a stronger foundation in geometry.


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