Calculator
Plotly graph
Example data
These examples show consistent results across different input modes.
| Method | Inputs | Minor segment area | Major segment area |
|---|---|---|---|
| Radius + angle | r=10 cm, θ=60° | 9.058 (cm²) | 305.101 (cm²) |
| Radius + chord | r=10 cm, c≈10.000 cm | 9.058 (cm²) | 305.101 (cm²) |
| Radius + height | r=10 cm, h≈1.340 cm | 9.058 (cm²) | 305.101 (cm²) |
Formula used
The calculator uses the standard circle segment relationship for the minor segment:
A_minor = (r²/2) · (θ − sin θ)A_sector = (r²/2) · θA_triangle = (r²/2) · sin θA_minor = A_sector − A_triangle
For the major segment:
A_major = πr² − A_minor
When you enter height or chord, the calculator first derives θ, then applies the same area formula.
How to use this calculator
- Select an input method: angle, height, or chord.
- Enter radius and your chosen secondary input.
- Choose degrees or radians when using an angle.
- Press Calculate to display areas above the form.
- Use Download CSV or Download PDF for exporting.
Why segment area matters
Circle segments appear in tanks, arches, lenses, and road curves. When you know the segment area, you can estimate fill volume per unit depth, compute cut‑out material, or price curved panels. For a radius of 10 cm and a 60° central angle, the minor segment is about 9.058 cm², while the remaining major part is about 305.101 cm².
Inputs that drive accuracy
The calculator accepts radius plus one extra measure: central angle, chord length, or segment height. Using consistent length units is essential because area scales with r². Doubling radius multiplies both minor and major areas by four. Small input errors can grow quickly, especially when the segment is thin and the angle is small.
Angle method performance
With θ in radians, the minor segment area is A = (r²/2)(θ − sinθ). This difference subtracts the isosceles triangle from the sector. When θ is near 0, A shrinks roughly with θ³, so more decimal precision helps. When θ approaches 2π, the minor segment approaches the full circle area.
Chord and height cross-checks
If you measure a chord c, the calculator derives θ = 2·asin(c/(2r)). If you measure height h (sagitta), it uses θ = 2·acos((r − h)/r). For r=10 cm, c≈10.000 cm and h≈1.340 cm both map to θ≈60°, producing the same minor area. These alternate paths are useful when angles are hard to measure directly.
Rounding, units, and reporting
Results are shown with six decimals to support engineering tolerances and classroom checking. Exported CSV includes raw numeric values for spreadsheets, while the PDF report lists the derived θ, sector area, triangle area, and both segment areas. Units are treated as labels, so the displayed cm² or m² depends on what you type.
Common validation limits
Geometric limits protect against impossible inputs: radius must be positive, chord must be at most 2r, and height must be less than 2r. The derived angle is restricted to (0, 2π]. If you need a small segment, prefer the angle method in radians and enter enough significant digits to avoid rounding artifacts.
FAQs
What is the difference between minor and major segment area?
The minor segment is the smaller region cut by a chord. The major segment is the remainder of the circle, computed as πr² minus the minor area.
Which input method should I choose?
Use angle when you know the central angle. Use chord when you can measure endpoints directly. Use height when you can measure sagitta from the chord to the arc.
Why does the calculator convert degrees to radians?
The area formula uses θ in radians for consistent trigonometric evaluation. Degrees are converted internally so you can enter angles in the unit you prefer.
Can this handle a semicircle segment?
Yes. Enter θ = 180° (or π radians). The minor segment becomes a semicircle, and the major segment matches the other half of the circle.
What if my chord is longer than 2r?
That input is geometrically impossible for a circle. The calculator blocks it and asks you to correct the radius or chord length.
Do the downloads include my units and intermediate values?
Yes. The CSV includes inputs, derived θ, and areas. The PDF report includes θ in degrees and radians plus sector and triangle areas used in the computation.