Area of Major Segment of a Circle — Formula & Concept

A segment is the region bounded by a chord and an arc of the circle.

• Minor Segment: The smaller region bounded by the chord and the minor arc.
• Major Segment: The larger region bounded by the same chord and the major arc.

(Don't confuse segment with sector. A sector is bounded by two radii and an arc, like a slice of pizza. A segment is bounded by a straight line chord and an arc).

There is no direct formula to find the area of the major segment. Instead, it is calculated by subtraction.

Area of Major Segment = Area of the whole Circle − Area of the Minor Segment

To break it down into steps:

1. Find the Area of the Circle = πr²
2. Find the Area of the Minor Segment = (Area of the Minor Sector) − (Area of the Triangle formed by the chord and radii)
3. Subtract the minor segment from the total circle area.

Problem: A chord of a circle of radius 10 cm subtends a right angle (90°) at the center. Find the area of the major segment. (Use π = 3.14)

Step 1: Area of the whole circle Area = πr² = 3.14 × 10² = 314 cm²

Step 2: Area of the minor segment Area of Minor Sector = (θ/360) × πr² = (90/360) × 314 = ¼ × 314 = 78.5 cm² Area of Triangle = ½ × base × height = ½ × 10 × 10 = 50 cm² Area of Minor Segment = 78.5 − 50 = 28.5 cm²

Step 3: Area of the major segment Area of Major Segment = Total Area − Minor Segment = 314 − 28.5 = 285.5 cm²