Circular Segment Area Calculator
Find the area of a circular segment — the region between a chord and its arc. Enter the radius plus the central angle, the chord length, or the sagitta (segment height). You also get the sector area, triangle area, arc length, and a live diagram with step-by-step working.
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Worked examples
Radius 10 cm, central angle 90°
r = 10 cm, θ = 90°
| Central angle | 90.0000° |
| Segment area | 28.5398 cm² |
| Sector area | 78.5398 cm² |
| Triangle area | 50.0000 cm² |
| Chord length | 14.1421 cm |
| Sagitta (height) | 2.9289 cm |
| Arc length | 15.7080 cm |
| Full circle area | 314.1593 cm² |
| Complementary segment | 285.6194 cm² |
Radius 10 cm, chord 8 cm
r = 10 cm, chord = 8 cm
| Central angle | 47.1564° |
| Segment area | 4.4911 cm² |
| Sector area | 41.1517 cm² |
| Triangle area | 36.6606 cm² |
| Chord length | 8.0000 cm |
| Sagitta (height) | 0.8348 cm |
| Arc length | 8.2303 cm |
| Full circle area | 314.1593 cm² |
| Complementary segment | 309.6682 cm² |
Radius 10 cm, sagitta 2 cm
r = 10 cm, sagitta = 2 cm
| Central angle | 73.7398° |
| Segment area | 16.3501 cm² |
| Sector area | 64.3501 cm² |
| Triangle area | 48.0000 cm² |
| Chord length | 12.0000 cm |
| Sagitta (height) | 2.0000 cm |
| Arc length | 12.8700 cm |
| Full circle area | 314.1593 cm² |
| Complementary segment | 297.8092 cm² |
Chord 8 cm and sagitta 2 cm (radius unknown)
chord = 8 cm, sagitta = 2 cm
Derived radius: r = (2² + 4²)/(2·2) = 5 cm.
| Central angle | 106.2602° |
| Segment area | 11.1824 cm² |
| Sector area | 23.1824 cm² |
| Triangle area | 12.0000 cm² |
| Chord length | 8.0000 cm |
| Sagitta (height) | 2.0000 cm |
| Arc length | 9.2730 cm |
| Full circle area | 78.5398 cm² |
| Complementary segment | 67.3574 cm² |
Radius 10 cm, central angle 270° (major segment)
r = 10 cm, θ = 270°
Signed triangle area -50.0000 cm² (θ > 180°): segment = sector − (-50.0000) = sector + 50.0000.
| Central angle | 270.0000° |
| Segment area | 285.6194 cm² |
| Sector area | 235.6194 cm² |
| Triangle area | 50.0000 cm² |
| Chord length | 14.1421 cm |
| Sagitta (height) | 17.0711 cm |
| Arc length | 47.1239 cm |
| Full circle area | 314.1593 cm² |
| Complementary segment | 28.5398 cm² |
Frequently asked questions
What is the area of a circular segment?
A circular segment is the region between a chord and its arc. Its area is A = (r²/2)·(θ − sinθ), with the central angle θ in radians. Example: radius 10 cm and central angle 90° → θ = 1.5708 rad, A = 50·(1.5708 − 1) = 28.5398 cm². The calculator above also shows the sector, triangle, chord, sagitta, and arc length.
How do you calculate segment area from chord length?
First recover the central angle: θ = 2·arcsin(c/(2r)). Example: radius 10 cm, chord 8 cm → θ = 2·arcsin(0.4) = 47.1564° = 0.8230 rad, then A = 50·(0.8230 − 0.7332) = 4.4911 cm². Choose the “Radius + Chord” mode above to do this automatically.
How do you calculate segment area from the sagitta (segment height)?
Recover the angle with θ = 2·arccos(1 − h/r). Example: radius 10 cm, sagitta 2 cm → θ = 2·arccos(0.8) = 73.7398°, A = 50·(1.2870 − 0.96) = 16.3501 cm². Use the “Radius + Sagitta” mode for this case.
What if I only know the chord and the segment height?
Recover the radius first: r = (h² + (c/2)²)/(2h). Example: chord 8 cm, sagitta 2 cm → r = (4 + 16)/4 = 5 cm, θ = 106.2602°, A = 11.1824 cm². The “Chord + Sagitta” mode handles exactly this situation.
What is the difference between a segment, a sector, and a triangle?
A sector is the pie slice bounded by two radii and the arc (area r²θ/2). The triangle is the isosceles triangle formed by the two radii and the chord (area r²·sinθ/2). The segment is what remains: sector − triangle. Example with r = 10 cm and θ = 90°: sector 78.5398 cm², triangle 50 cm², segment 78.5398 − 50 = 28.5398 cm².
How do you find the area of a major segment (more than a semicircle)?
Use a central angle above 180° — the same formula A = (r²/2)(θ − sinθ) still works, or subtract the minor segment from the full circle: πr² − minor. Example: r = 10 cm, θ = 270° → A = 50·(4.7124 + 1) = 285.6194 cm²; the complementary minor segment is 28.5398 cm² and the full circle is 314.1593 cm².
How do you calculate the arc length of a segment?
Arc length L = r·θ with θ in radians. Example: r = 10 cm, θ = 90° → L = 10·1.5708 = 15.7080 cm. The calculator reports the arc length for every input mode.
What is the difference between this tool and a circle area calculator?
A circle area calculator computes the area of a FULL circle (A = πr²) only. This tool computes the area of a circular SEGMENT — just the cap between a chord and its arc — plus the sector, triangle, chord, sagitta, and arc length. A semicircle is simply the θ = 180° special case (area πr²/2 = 157.0796 cm² for r = 10 cm).
Related tools
Circle Area Calculator — area of a full circle from radius, diameter, or circumference.
Circle Dimension Hub — convert between circumference, diameter, radius, and area.
Free tool by NRTechStudio