Segment and sector of a circle

You are familiar with some methods of finding the area of plane figures like triangle, square, rectangle, parallelogram, rhombus, and of the circle. Here the area of segment and sector of the circle is of great importance in our day to life like areas of the rounded figure as an example, wheel excel, and other rounded tools and machines. The segment and sector are the part of the rounded figure are needed in manufacturing different types of tools used in machines and architectural designs of bridges and buildings, so the area of segments and sectors of the circle is introduced in the 10-grade syllabus of every school board.

NCERT Solutions Class 10 Science from chapter 1 to 16

Area of the segment and sectors is the topic referred from chapter 12 of the class 10 NCERT book prescribed by CBSE a reputed board of India for school education.

Sector of circle :

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The area covered by two radii and the corresponding arc of the circle is known as sector.

360° angle around the centre corresponds to the area $= \pi r^{2}$

1° angle around the circle corresponds to the area $= \frac{\pi r^{2}}{360}$

θ angle around the circle correspond to the area $= \frac{\pi r^{2}}{360}\times \Theta$

Hence the area of a sector

$= \frac{\Theta }{360}\times \pi r^{2}$

Where θ is the angle between two radii of the circle.

Type of sectors : Sectors of circles are of two kinds (a) Minor sector (b) Major sector

Minor sector: The shaded region in above figure is minor sector i.e The smaller sector is known as minor sector.

Major sector: The unshaded region in above figure is major sector i.e The larger sector is known as major sector.

Area of major sector = Area of circle – area of minor sector

$Area \: of\: major \: sector =\pi r^{2}-\frac{\Theta }{360}\pi r^{2}$

Segment of a circle :

The area covered by the chord of a circle and corresponding arc known as segment.

Area of segment = area of sector – area of triangle covered by two radii and corresponding chord of the circle

Let the angle subtended by the two radii is = θ

∠AOM = θ/2 (OM⊥ AB)

$sin \frac{\Theta }{2} =\frac{AM}{r}$

AM = r sinθ/2

AB = 2AM=2r sinθ

$cos \frac{\Theta }{2} =\frac{OM}{r}$

OM = r cosθ/2

Area of ΔAOB

$=\frac{1}{2}AB\times OM$

$=\frac{1}{2}2rsin\frac{\Theta }{2}.rcos\frac{\Theta }{2}$

Since ,2sin θ cos θ =sin2θ

$=\frac{1}{2}r^{2}sin\Theta$

Hence area of ΔAOB = 1/2 r² sinθ

We know area of sector

$= \frac{\Theta }{360}\times \pi r^{2}$

So, area of segment

$=\frac{\Theta }{360}\pi r^{2}-\frac{1}{2}r^{2}sin\Theta$

Hence Area of segment

$=\frac{r^{2}}{2}\left ( \frac{\pi \Theta }{180}-sin\Theta \right )$

Type of segment

(a) Minor segment: The smaller segment of the circle is known as minor segment i.e The shaded region in above figure is known as minor segment.

(b) Major segment: The larger segment of the circle is known as major segment i.e The unshaded region in above figure is known as major segment.

The area of larger segment = Area of circle – area of minor sector

$Area \: of \: major \: segment=\pi r^{2}-\frac{r^{2}}{2}\left ( \frac{\pi \Theta }{180}-sin\Theta \right )$

Length of arc :

Let the length of arc corresponding to two radii of circle is l.

360° angle corresponds to $=2\pi r$

1° angle corresponds to $=\frac{2\pi r}{360}$

θ angle corresponds to $=\frac{2\pi r}{360}\times \Theta$

Hence the length of arc corresponding to two radii $l= \frac{\Theta }{360} \times 2\pi r$

Type of Arc

(a) Minor arc: The smaller arc of the circle is known as minor arc, in the figure l is the length of minor arc.

$l=\frac{\Theta }{360^{0}}\times 2\pi r$

(b) Major arc: The larger arc of the circle is known as major arc

The length of major arc = Circumference of the circle – The length of minor arc

$The \: length \: of \: major\: arc=2\pi r-\frac{\Theta }{360}\times 2\pi r$

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NCERT Solutions of Science and Maths for Class 9,10,11 and 12

NCERT Solutions for class 9 maths

Chapter 1- Number System	Chapter 9-Areas of parallelogram and triangles
Chapter 2-Polynomial	Chapter 10-Circles
Chapter 3- Coordinate Geometry	Chapter 11-Construction
Chapter 4- Linear equations in two variables	Chapter 12-Heron’s Formula
Chapter 5- Introduction to Euclid’s Geometry	Chapter 13-Surface Areas and Volumes
Chapter 6-Lines and Angles	Chapter 14-Statistics
Chapter 7-Triangles	Chapter 15-Probability
Chapter 8- Quadrilateral

NCERT Solutions for class 9 science

Chapter 1-Matter in our surroundings	Chapter 9- Force and laws of motion
Chapter 2-Is matter around us pure?	Chapter 10- Gravitation
Chapter3- Atoms and Molecules	Chapter 11- Work and Energy
Chapter 4-Structure of the Atom	Chapter 12- Sound
Chapter 5-Fundamental unit of life	Chapter 13-Why do we fall ill ?
Chapter 6- Tissues	Chapter 14- Natural Resources
Chapter 7- Diversity in living organism	Chapter 15-Improvement in food resources
Chapter 8- Motion	Last years question papers & sample papers

NCERT Solutions for class 10 maths

Chapter 1-Real number	Chapter 9-Some application of Trigonometry
Chapter 2-Polynomial	Chapter 10-Circles
Chapter 3-Linear equations	Chapter 11- Construction
Chapter 4- Quadratic equations	Chapter 12-Area related to circle
Chapter 5-Arithmetic Progression	Chapter 13-Surface areas and Volume
Chapter 6-Triangle	Chapter 14-Statistics
Chapter 7- Co-ordinate geometry	Chapter 15-Probability
Chapter 8-Trigonometry

CBSE Class 10-Question paper of maths 2021 with solutions

CBSE Class 10-Half yearly question paper of maths 2020 with solutions

CBSE Class 10 -Question paper of maths 2020 with solutions

CBSE Class 10-Question paper of maths 2019 with solutions

NCERT Solutions for Class 10 Science

Chapter 1- Chemical reactions and equations	Chapter 9- Heredity and Evolution
Chapter 2- Acid, Base and Salt	Chapter 10- Light reflection and refraction
Chapter 3- Metals and Non-Metals	Chapter 11- Human eye and colorful world
Chapter 4- Carbon and its Compounds	Chapter 12- Electricity
Chapter 5-Periodic classification of elements	Chapter 13-Magnetic effect of electric current
Chapter 6- Life Process	Chapter 14-Sources of Energy
Chapter 7-Control and Coordination	Chapter 15-Environment
Chapter 8- How do organisms reproduce?	Chapter 16-Management of Natural Resources

NCERT Solutions for class 11 maths

Chapter 1-Sets	Chapter 9-Sequences and Series
Chapter 2- Relations and functions	Chapter 10- Straight Lines
Chapter 3- Trigonometry	Chapter 11-Conic Sections
Chapter 4-Principle of mathematical induction	Chapter 12-Introduction to three Dimensional Geometry
Chapter 5-Complex numbers	Chapter 13- Limits and Derivatives
Chapter 6- Linear Inequalities	Chapter 14-Mathematical Reasoning
Chapter 7- Permutations and Combinations	Chapter 15- Statistics
Chapter 8- Binomial Theorem	Chapter 16- Probability

CBSE Class 11-Question paper of maths 2015

CBSE Class 11 – Second unit test of maths 2021 with solutions

NCERT solutions for class 12 maths

Chapter 1-Relations and Functions	Chapter 9-Differential Equations
Chapter 2-Inverse Trigonometric Functions	Chapter 10-Vector Algebra
Chapter 3-Matrices	Chapter 11 – Three Dimensional Geometry
Chapter 4-Determinants	Chapter 12-Linear Programming
Chapter 5- Continuity and Differentiability	Chapter 13-Probability
Chapter 6- Application of Derivation	CBSE Class 12- Question paper of maths 2021 with solutions
Chapter 7- Integrals
Chapter 8-Application of Integrals