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Area Related to Circles Class 10 Notes Maths Chapter 11

  1. Circumference of a circle = 2πr
  2. Area of a circle = πr2 …[where r is the radius of a circle]
  3. Area of a semi-circle =πr2/2
  4. Area of a circular path or ring: 
    Area Related to Circles Class 10 Notes Maths Chapter 11Let ‘R’ and ‘r’ he radii of two circles 
    Then area of shaded part = πR2 – πr2 = π(R2 – r2) = π(R + r)(R – r) 

Minor Arc and Major Arc

An arc length is called a major arc if the arc length enclosed by the two radii is greater than a semi-circle.

Area Related to Circles Class 10 Notes Maths Chapter 11
If the arc subtends angle ‘θ’ at the centre, then the 
Length of minor arc = Area Related to Circles Class 10 Notes Maths Chapter 11
Length of major arc = Area Related to Circles Class 10 Notes Maths Chapter 11

Sector of a Circle and its Area

A region of a circle is enclosed by any two radii and the arc intercepted between two radii is called the sector of a circle.

Minor Sector

A sector is called a minor sector if the minor arc of the circle is part of its boundary.
OAPB is minor sector.

Area Related to Circles Class 10 Notes Maths Chapter 11Area of minor sector = Area Related to Circles Class 10 Notes Maths Chapter 11
Perimeter of minor sector = Area Related to Circles Class 10 Notes Maths Chapter 11

Major Sector

A sector is called a major sector if the major arc of the circle is part of its boundary. 
OAQB is major sector
Area of major sector = Area Related to Circles Class 10 Notes Maths Chapter 11
Perimeter of major sector = Area Related to Circles Class 10 Notes Maths Chapter 11

Question for Short Notes: Areas Related to Circles
Try yourself:An arc length is called a major arc if:
View Solution

Segment of a Circle and its Area

Minor Segment

The region enclosed by an arc and a chord is called a segment of the circle. The region enclosed by the chord AB & minor arc ACB is called the minor segment. 

Area Related to Circles Class 10 Notes Maths Chapter 11
Area of Minor segment = Area of the corresponding sector – Area of the corresponding triangle
Area Related to Circles Class 10 Notes Maths Chapter 11

Major Segment

The region enclosed by the chord AB & major arc ADB is called the major segment. Area of major segment = Area of a circle – Area of the minor segment
Area of major sector + Area of triangle
Area Related to Circles Class 10 Notes Maths Chapter 11

Example: A chord of a circle of radius 12 cm subtends an angle of 120° at the centre. Find the area of the corresponding segment of the circle. (Use π = 3.14 and √3 = 1.73).

Sol:

We use the concept of areas of sectors of circles to solve the question.

In a circle with radius r and the angle at the centre with degree measure θ:

(i) Area of the sector = θ/360 πr2

(ii) Area of the segment = Area of the sector - Area of the corresponding triangle

Let's draw a figure to visualize the given question.

Area Related to Circles Class 10 Notes Maths Chapter 11

Here, radius, r = 12 cm, ∠AOB = θ = 120°

Visually it’s clear from the figure that AB is the chord that subtends 120° angle at the centre.

To find the area of the segment AYB, we have to find the area of the sector OAYB and the area of the ΔAOB

(i) Area of sector OAYB = θ/360° πr2

(ii) Area of ΔAOB = 1/2 × base × height

For finding the area of ΔAOB, draw OM ⊥ AB then find base AB and height OM using the figure as shown above.

Area of sector OAYB = 120°/360° × πr2

= 1/3 × 3.14 × (12 cm)2

= 150.72 cm2

Draw a perpendicular OM from O to chord AB

In ΔAOM and ΔBOM

AO = BO = r (radii of circle)

OM = OM (common side)

∠OMA = ∠OMB = 90° (perpendicular OM drawn)

∴ ΔAOM ≅ ΔBOM (By RHS Congruency)

⇒ ∠AOM = ∠BOM (By CPCT)

Therefore, ∠AOM = ∠BOM = 1/2 ∠AOB = 60°

In ΔAOM,

AM/OA = sin 60° and OM/OA = cos 60°

AM/12 cm = √3/2 and OM/12 cm = 1/2

AM = √3/2 × 12 cm and OM = 1/2 × 12 cm

AM = 6√3 cm and OM = 6 cm

⇒ AB = 2 AM

= 2 × 6√3 cm

= 12√3 cm

Area of ΔAOB = 1/2 × AB × OM

= 1/2 × 12√3 cm × 6 cm

= 36 × 1.73 cm2

= 62.28 cm2

Area of segment AYB = Area of sector OAYB - Area of ΔAOB

= 150.72 cm2 - 62.28 cm2

= 88.44 cm2

Question for Short Notes: Areas Related to Circles
Try yourself:What is the area of the region enclosed by a chord AB and the minor arc ACB of a circle called?

 

Area Related to Circles Class 10 Notes Maths Chapter 11

View Solution

Table for Area and Perimeter of Circle

Area Related to Circles Class 10 Notes Maths Chapter 11


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