Library / Geometry, coordinates, surveying and trigonometry / GEO-08
Geometry, coordinates, surveying and trigonometry
Area of a circular segment
Area between a chord and its arc, from the radius and the angle the arc subtends.
- Ref
- GEO-08
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- A [cm²]
The segment is the part of a disc cut off by a chord. Its area is the area of the sector, ½r²θ, minus the area of the triangle, ½r² sin θ. This gives ½r²(θ − sin θ). For angles above π the same formula gives the major segment. Below about 0.01 rad the difference θ − sin θ is replaced by θ³/6 − θ⁵/120 to avoid loss of digits.
Area between a chord and its arc, from the radius and the angle the arc subtends. Values above π give the major segment. For tiny angles a series is used so that the subtraction does not lose digits.
Calculator
- Area of the sector, r²θ/212.57 cm²
- Area of the triangle, r² sin θ / 28 cm²
- Segment as a share of the whole disc0.09085
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A 90° arc on a 4 cm radius cuts off 4.566 cm², which is 9.08 % of the whole disc.
- For an arc of π radians (a half disc) the segment is half the disc, π r²/2.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| r Radius | 1 to 20 cm | 0.2854 | 114.2 | 113.9 | +10% | +21 % |
| θ Angle subtended by the arc | 0.05 to 6.283 rad | 1.666e-4 | 50.27 | 50.27 | +10% | +29.7 % |
Sweep of Angle subtended by the arc
| Angle subtended by the arc (rad) | Segment area (cm²) |
|---|---|
| 0.05 | 1.666e-4 |
| 0.8291 | 0.7343 |
| 1.608 | 4.872 |
| 2.387 | 13.62 |
| 3.167 | 25.53 |
| 3.946 | 37.33 |
| 4.725 | 45.8 |
| 5.504 | 49.65 |
| 6.283 | 50.27 |
Sweep one variable
Variables and how each one contributes
Segment area
What the equation gives you. Everything else feeds it.
Radius
The radius of the circle, in cm. A larger radius gives a segment of larger area with the square of the radius. Raising it by 10 % (to 4.4) moves the result by +21 %, from 4.566 to 5.525 cm².
Angle subtended by the arc
The angle subtended at the centre by the arc, in radians, from 0 to 2π. A wider angle gives a much larger area, and the area grows faster than the angle. Raising it by 0.05 (to 1.621) moves the result by +8.98 %, from 4.566 to 4.976 cm².
Worked example with the values above
- Sector: ½r²θ = ½ × 16 × 1.571 = 12.566 cm².
- Triangle: ½r² sin θ = 8 cm².
- Segment = 12.566 − 8 = 4.5664 cm².
Assumptions and limits
- A true circle, with angles in radians where the formula uses them.
- Lengths are measured from the same centre, in one consistent unit.
Common mistakes
- Entering a central angle in degrees where radians are needed.
- Using the full chord in place of the half chord when working with the right triangle.
- Using the major arc when the minor arc is meant.
Statistics and uncertainty
Circle results depend on the radius squared or on trigonometric functions of the angle, so errors in the radius are doubled in areas. The segment and rise formulas can lose digits for very small angles or very shallow arcs.
Where it is used
- Cutting, rolling and setting out curved plates, pipes and arches.
- Partly filled pipes and tanks, washers and flanges, and tangent lines past a circular object.
Maths you need
- Radians: angle measured as arc length over radius
- Right triangle: Pythagoras and sine over hypotenuse
- Circle area: π times radius squared