A circle is the set of all points in a plane which are at the
same distance from a fixed point called the centre.
That fixed distance is called the radius.
Circumference = 2πr
Area = πr²
Diameter = 2r
💡 Circle ki boundary ki total length = circumference.
Circle ke andar covered region = area.
02. Circular Path / Ring
When two concentric circles form a region between them,
that region is called a circular path.
Let:
R = outer radius
r = inner radius
Area of Circular Path
=
πR² − πr²
= π(R² − r²)
✏️ Example
Outer radius = 7 cm
Inner radius = 3 cm
Area = π(7² − 3²)
= π(49 − 9)
= 40π cm²
✓ Area = 40π cm²
03. Sector of a Circle ⭐⭐⭐⭐⭐
A sector is the region enclosed by:
Two radii + the corresponding arc.
The angle between the two radii is called the
central angle.
💡 Sector ko identify karne ka easiest way:
Centre se do radii draw hongi aur unke beech arc hoga.
04. Minor Sector — Formula ⭐⭐⭐⭐⭐
The sector corresponding to the smaller central angle is called
the minor sector.
Usually:
θ < 180°
Area of Minor Sector
=
θ360°
× πr²
Arc Length of Minor Sector
=
θ360°
× 2πr
✏️ Example
Find the area of a sector of radius 14 cm and angle 90°.
Area
=
90°360°
× π × 14²
=
14
× π × 196
Using π =
227:
= 154 cm²
✓ Area of Minor Sector = 154 cm²
05. Major Sector — Formula ⭐⭐⭐⭐⭐
The larger region formed by the two radii and the major arc is
called the major sector.
If the minor angle is θ:
Major Angle
= 360° − θ
Area of Major Sector
=
πr² − Area of Minor Sector
=
πr² −
[
θ360°
× πr²
]
=
360° − θ360°
× πr²
✏️ Example
A circle has radius 14 cm and the minor sector angle is 90°.
Find the area of the major sector.
Major angle = 360° − 90°
= 270°
Area
=
270°360°
× π × 14²
=
34
× 616
= 462 cm²
✓ Area of Major Sector = 462 cm²
06. Sector — Complete Formula Box
① Minor Sector
Area =
θ360°
πr²
Arc Length =
θ360°
2πr
② Major Sector
Major Angle = 360° − θ
Area =
360° − θ360°
πr²
OR
Area = πr² − Minor Sector Area
🧠 Sector Trick:
Minor → θ/360 × πr²
Major → (360 − θ)/360 × πr²
Full Circle → πr²
07. Segment of a Circle ⭐⭐⭐⭐⭐
A segment is the region bounded by a chord and its
corresponding arc.
Sector → Two radii + Arc Segment → Chord + Arc
08. Minor Segment — Formula ⭐⭐⭐⭐⭐
Area of Minor Segment
= Area of Minor Sector
− Area of ΔOAB
=
θ360°
πr²
− Area of ΔOAB
💡 Segment ka direct circle formula yaad karne ki zarurat nahi.
Pehle sector ka area nikalo.
Phir triangle OAB ka area subtract karo.
Sector − Triangle = Minor Segment
✏️ Example
For a circle of radius 7 cm and central angle 90°, find the
area of the minor segment.
Sector area:
=
90°360°
× π × 7²
= 38.5 cm²
Triangle area:
=
12
× 7 × 7
= 24.5 cm²
Therefore:
Minor Segment
= 38.5 − 24.5
= 14 cm²
✓ Area of Minor Segment = 14 cm²
09. Major Segment — Formula ⭐⭐⭐⭐⭐
The larger region of a circle bounded by a chord and the major
arc is called the major segment.
Area of Major Segment
= Area of Circle − Area of Minor Segment
= πr² − Area of Minor Segment
= πr² −
[
θ360°
πr² − Area of ΔOAB
]
🧠 Major segment ko directly solve karne ke bajay usually:
Full Circle − Minor Segment
karna easiest hota hai.
✏️ Example
If the area of the circle is 154 cm² and the area of its
minor segment is 14 cm²:
Major Segment
= 154 − 14
= 140 cm²
✓ Area of Major Segment = 140 cm²
10. Major vs Minor — Quick Comparison
Concept
Minor
Major
Sector
Smaller sector
Larger sector
Angle
θ
360° − θ
Sector Area
θ360°
πr²
360° − θ360°
πr²
Segment
Smaller region
Larger region
Segment Area
Sector − Triangle
Circle − Minor Segment
11. Arc Length ⭐⭐⭐⭐⭐
An arc is a part of the circumference of a circle.
Length of Arc
=
θ360°
× 2πr
✏️ Example
Find the arc length of a circle with radius 14 cm and central
angle 90°.
=
90°360°
× 2π × 14
=
14
× 28π
Using π = 22/7:
= 22 cm
✓ Arc Length = 22 cm
12. Area of Sector + Triangle Figure
Exam Identification:
If the question asks for the shaded region between a chord and
an arc, think:
Segment
If the question asks for a region between two radii and an arc,
think:
Sector
13. Shaded Region Strategy ⭐⭐⭐⭐⭐
Most shaded-area questions can be solved using:
Required Region = Bigger Region − Unwanted Region
Circle − Inner Circle
Circle − Sector
Sector − Triangle
Outer Circle − Inner Circle
Circle − Minor Segment
🧠 Figure dekhte hi calculation start mat karo.
Pehle identify karo:
“Shaded region actually kis shape ka part hai?”
14. Circular Path — Exam Example
Question
A circular garden has radius 14 m. A path of width 3.5 m is
constructed inside it. Find the area of the path.
Outer radius:
R = 14 m
Inner radius:
r = 14 − 3.5
r = 10.5 m
Therefore:
Area
=
π(14² − 10.5²)
Using π = 22/7:
= 269.5 m²
✓ Area of Path = 269.5 m²
15. Special Angles — Easy Fraction Connection
Angle
Fraction of Circle
Sector Area
90°
14
14
πr²
180°
12
12
πr²
270°
34
34
πr²
360°
1
πr²
16. Complete Formula Sheet ⭐⭐⭐⭐⭐
Diameter = 2r
Circumference = 2πr
Area of Circle = πr²
Area of Minor Sector
=
θ360°
πr²
Area of Major Sector
=
360° − θ360°
πr²
Arc Length
=
θ360°
2πr
Minor Segment
=
Minor Sector − Triangle
Major Segment
=
Circle − Minor Segment
Circular Path
=
π(R² − r²)
17. Sector vs Segment — Never Confuse
Sector
Segment
Two radii + arc
Chord + arc
Centre is used
Chord is used
θ360°
πr²
Sector − Triangle
❤️ Ek line yaad rakho:
SECTOR → Radius + Radius + Arc
SEGMENT → Chord + Arc
18. Common Exam Mistakes ⚠️
1. Diameter ko radius maan lena.
2. Minor aur major sector confuse karna.
3. Sector aur segment ko same samajhna.
4. θ/360° ko incorrectly use karna.
5. Circular path mein inner area subtract na karna.