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.
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
1° angle around the circle corresponds to the area
θ angle around the circle correspond to the area
Hence the area of a sector
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
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)
AM = r sinθ/2
AB = 2AM=2r sinθ
OM = r cosθ/2
Area of ΔAOB
Since ,2sin θ cos θ =sin2θ
Hence area of ΔAOB = 1/2 r² sinθ
We know area of sector
So, area of segment
Hence Area of segment
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
Length of arc :
Let the length of arc corresponding to two radii of circle is l.
360° angle corresponds to
1° angle corresponds to
θ angle corresponds to
Hence the length of arc corresponding to two radii
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.
(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
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|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|
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|
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