Area of Segment with Sector and Triangle Calculations
Area of Segment is the region enclosed by a chord and its corresponding arc of a circle. Its area is found by subtracting the area of the triangle formed by the two radii and chord from the sector area. This page covers minor and major segments, formulas in degrees and radians, solved examples and useful calculation rules.
What Is the Area of a Segment?
If the radius is r and the angle subtended by the chord at the centre is θ, the minor segment is obtained from the sector OAB by removing triangle OAB. When θ is measured in degrees, use the degree formula. The major segment is the remaining part of the circle after removing the minor segment.
Area of Segment Video Lessons
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Circle Sector and Segment Area
Learn how to calculate the areas of sectors and segments of a circle using their standard formulas and the given radius, angle, or related measurements.
Area of Segment Formula & Tricks
Important Formulas
Here, θ is the central angle in degrees. The first term is the sector area and the second term is the area of the included triangle.
Use this form when θ is measured in radians.
The major segment and minor segment together form the complete circle.
This is the part of the circle bounded by two radii and the included arc.
The two sides enclosing θ are radii of length r.
Quick Tricks
For a minor segment, first calculate the sector area and then subtract the isosceles triangle area. This avoids confusing a segment with a sector.
Calculate the minor segment first, then subtract it from πr². Direct calculation of the larger curved region is usually longer.
When θ = 180°, the chord is a diameter and each segment is a semicircle. The segment area is πr²/2.
Area of Segment Concepts
Minor Segment Formula in Degrees
The central angle θ must be in degrees in this formula. For θ less than 180°, the smaller region cut off by the chord is the minor segment. The sine value uses the same degree measure.
Major Segment Area
The full circle area is πr². If the given central angle is greater than 180°, it describes the major segment directly, but it is usually simpler to use the supplementary minor sector angle, 360° − θ, and then subtract the resulting minor segment from the circle.
Segment Area in Radians
The sector area in radians is (1/2)r²θ. Subtracting the triangle area, (1/2)r² sin θ, gives the stated formula. Convert degrees to radians using θ radians = θ° × π/180 before applying it.
Difference Between a Sector and a Segment
Sector area uses only (θ/360)πr² in degrees. Segment area requires the triangle subtraction because the straight boundary is a chord rather than two radii.
Area of Segment Visual Guide & Data
Use the table and chart below to understand the data better.
Comparison of Circle Regions
The boundaries and area formulas distinguish a sector, minor segment and major segment.
| Region | Boundary or construction | Area formula |
|---|---|---|
| Sector | Two radii and the included arc | (θ/360)πr² |
| Minor segment | Chord and minor arc | (θ/360)πr² − (1/2)r² sin θ |
| Major segment | Major arc and the same chord | πr² − minor segment area |
Note: For the segment formulas, θ is the central angle in degrees.
Minor Segment Formed by a Chord
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Area of Segment Revision Points
Use these formulas and rules to revise segment-area calculations.
- A segment is bounded by a chord and an arc.
- Minor segment area = sector area − triangle area.
- In degrees: Area = (θ/360)πr² − (1/2)r² sin θ.
- In radians: Area = (1/2)r²(θ − sin θ).
- Major segment area = πr² − minor segment area.
- For θ = 180°, the segment is a semicircle with area πr²/2.
- Do not use the sector formula alone for a segment.
Area of Segment FAQs
What is the area of segment formula when the angle is in degrees?
Area of minor segment = (θ/360)πr² − (1/2)r² sin θ, where r is the radius and θ is the central angle in degrees.
How do you find the area of a major segment?
Subtract the minor segment area from the circle area: Area of major segment = πr² − Area of minor segment.
What is the area of a segment when the central angle is 180°?
The segment is a semicircle, so its area is πr²/2. For r = 7 cm, the area is 77 cm² using π = 22/7.
What is the difference between sector area and segment area?
Sector area is (θ/360)πr² and is bounded by two radii and an arc. Segment area is sector area minus (1/2)r² sin θ because its straight boundary is a chord.
Find the area of a minor segment with radius 7 cm and central angle 90°.
Area = (90/360)π(7²) − (1/2)(7²)sin 90° = 38.5 − 24.5 = 14 cm², using π = 22/7.
How is the segment formula used when the angle is given in radians?
Use Area = (1/2)r²(θ − sin θ). For r = 2 cm and θ = π/2, the area is π − 2 cm².