Mathematics: Class 10 — Chapter 11: Areas Related to Circles

1. Core Definitions

Sector of a circle: The portion of the circular region enclosed by two radii and the corresponding arc.
- Minor Sector: Sector with angle < 180°.
- Major Sector: Sector with angle 360° - θ.
Segment of a circle: The portion of the circular region enclosed between a chord and the corresponding arc.
- Minor Segment: Formed by the minor arc.
- Major Segment: Formed by the major arc.
Remark: Unless stated otherwise, segment and sector refer to the minor segment and minor sector.

Length of an arc of a sector (angle θ, radius r): (θ / 360) × 2πr
Area of a sector (angle θ, radius r): (θ / 360) × πr²
Area of a segment: Area of the corresponding sector - Area of the corresponding triangle

Example 1: Area of Sector

Given: Radius r = 4 cm, angle θ = 30°, π = 3.14.

Area of minor sector: (30/360) × 3.14 × 4² ≈ 4.19 cm²
Area of major sector: πr² - Area of minor sector = (3.14 × 16) - 4.19 ≈ 46.1 cm²

Example 2: Area of Segment

Given: Radius r = 21 cm, angle θ = 120°, π = 22/7.

Area of sector: (120/360) × (22/7) × 21² = 462 cm²
Area of triangle: Uses trigonometry to find base and height. Area = (441/4)√3 cm²
Area of segment: 462 - (441/4)√3 cm²

Common problem types involving sector and segment areas:

Calculating area swept by a clock's minute hand over a specific time.
Finding grazing area for tethered animals (e.g., horse tied to a square field corner).
Calculating material areas for symmetrical objects (e.g., brooches, umbrellas, round table covers).
Determining coverage area for sweeping mechanisms (e.g., car wipers, lighthouse beams).