Sectors and Segments of a Circle
1.0Master Sectors, Segments, and Circular Area Calculations in Minutes
Explore how circles are divided into sectors and segments and learn to calculate their areas with ease. Understand the relationship between central angles, arc lengths, and enclosed regions while solving practical problems involving gardens, roads, wheels, sports tracks, and architectural designs. With step-by-step calculations and board-oriented examples, this topic makes geometry simple and engaging.
2.0Learning Outcomes
After completing this topic, you will be able to:
- Define a sector and a segment of a circle.
- Differentiate between major and minor sectors.
- Differentiate between major and minor segments.
- Calculate the area of a sector.
- Find the length of an arc.
- Calculate the area of a segment.
- Solve real-life problems involving sectors and segments.
- Solve NCERT, competency-based, and CBSE Board examination questions confidently.
3.0Introduction
A circle can be divided into different regions using its radii and chords. Two such important regions are the sector and the segment of a circle. Understanding these concepts helps in solving problems related to the area of circular regions, arc lengths, and practical applications involving wheels, parks, pizza slices, clock faces, and engineering designs.
In this topic, you'll learn how to calculate the area of a sector, length of an arc, area of a segment, and solve application-based problems using the properties of circles. These concepts are frequently tested in the CBSE Class 10 Mathematics Board Examination.
4.0What are Sectors and Segments of a Circle?
A circle can be divided into different regions using radii and chords. Two of the most important regions are the sector and the segment. These concepts help us calculate the area of parts of a circle and solve real-life problems involving wheels, pizza slices, parks, clock faces, and circular tracks.
In Class 10 Mathematics, students learn how to calculate the area of a sector, length of an arc, and area of a segment using standard formulas. These concepts are an important part of the CBSE syllabus and frequently appear in board examinations.
5.0What is a Sector of a Circle?
A sector is the region enclosed by two radii of a circle and the arc between them.
Depending on the central angle, sectors are of two types:
- Minor Sector – Formed by the smaller arc (central angle less than 180°).
- Major Sector – Formed by the larger arc (central angle greater than 180°).
Examples of a Sector
- A slice of pizza
- A piece of cake
- A section of a circular garden
- A clock divided into hourly sections
6.0What is a Segment of a Circle?
A segment is the region enclosed by a chord and the arc corresponding to that chord. Segments are also classified into:
- Minor Segment
- Major Segment
Unlike a sector, a segment is not bounded by two radii.
7.0Difference Between Sector and Segment
8.0Area of a Sector
The area of a sector depends on the radius of the circle and the central angle.
Formula: Area of Sector = (θ / 360) × πr²
where:
θ = Central angle
r = Radius of the circle
Example
Find the area of a sector of radius 14 cm and central angle 90°.
Area
= (90/360) × (22/7) × 14²
= 154 cm²
9.0Length of an Arc
The curved boundary of a sector is known as the arc.
Formula: Length of Arc = (θ / 360) × 2πr
where:
- θ = Central angle
- r = Radius
Example
Find the arc length when
Radius = 21 cm
Central Angle = 60°
Length
= (60/360) × 2 × (22/7) × 21
= 22 cm
10.0Area of a Segment
The area of a segment is obtained by subtracting the area of the triangle formed by the two radii from the area of the corresponding sector.
Formula: Area of Segment = Area of Sector − Area of Triangle
Example
Radius = 14 cm
Central Angle = 90°
Area of Sector = 154 cm²
Area of Triangle = 1/2 × 14 × 14 = 98 cm²
Area of Segment
= 154 − 98 = 56 cm²
11.0Major and Minor Sectors
- A minor sector is formed when the central angle is less than 180°.
- A major sector is formed when the central angle is greater than 180°.
- The sum of the areas of the major and minor sectors equals the area of the complete circle.
12.0Major and Minor Segments
Similarly,
- The minor segment is enclosed by the smaller arc.
- The major segment is enclosed by the larger arc.
Together, they make up the complete area of the circle.
13.0Applications of Sectors and Segments
The concepts of sectors and segments are widely used in everyday life and engineering.
Some common applications include:
- Designing circular parks and fountains
- Measuring pizza and cake slices
- Constructing highways and flyovers
- Manufacturing gears and machine parts
- Designing sports stadiums
- Architecture and civil engineering
- Irrigation systems
- Clock and wheel designs
14.0Important Formulas