Area of minor and major segment of a circle

Area of Minor and Major Segment of a Circle

In geometry, a segment of a circle is the region bounded by a chord and the corresponding arc lying between the chord's endpoints. A circle segment can either be a minor segment or a major segment, depending on whether it covers less or more than half of the circle, respectively.

Key Terms

• Radius (r): The length from the circle’s center to any point along its boundary.
• Central Angle (θ): The angle formed at the circle’s center by lines extending to the ends of an arc, measured in degrees or radians.
• Chord: A straight line connecting two points on the circle.

Formulas

The area of a segment of a circle is calculated using the formula:

   Segment Area = Sector Area − Triangle Area
  

1. Area of Sector:

If θ is in degrees:
  Sector Area = (θ / 360) × π × r²

If θ is in radians:
  Sector Area = (1/2) × r² × θ
  

2. Area of Triangle (formed by two radii and the chord):

Area = (1/2) × r² × sin(θ)     // θ in radians
  

Minor and Major Segments

- The minor segment refers to the smaller portion of the circle enclosed between a chord and its corresponding arc.
- The major segment is the larger region, which is the rest of the circle outside the minor segment.

Hence:

Minor Segment Area = Sector Area − Area of Triangle
Area of Major Segment = π × r² - Area of Minor Segment
  

Example

Given a circle with radius r = 10 units and central angle θ = 60°:

1. Sector area can be calculated as (60 ÷ 360) × π × (10²), which simplifies to (1⁄6) × π × 100 = (50⁄3)π ≈ 52.36 square units.
2. Triangle Area = (1/2) × 10² × sin(60°) ≈ 43.30 units²
3. Approximate area of the minor segment: 52.36 units² (sector) minus 43.30 units² (triangle), resulting in about 9.06 units².
4. To find the major segment area: subtract the minor segment area from the total circle area (π × 100 ≈ 314.16), giving roughly 305.10 units².

The area of a segment of a circle is an important geometric concept used in engineering, architecture, and various fields of science. Understanding how to calculate both the minor and major segments allows for precise measurements in circular designs.