Sector and Segment
Find the arc length, sector area, chord, height and area of a circular segment from the radius and angle, the arc length or the chord.
The circular sector and segment calculator finds the arc length, the sector area, the chord, the height and the area of the segment. A sector can be given by the radius and the central angle, the radius and the arc length, or the radius and the chord.
Sector and segment
A sector is the part of a disk between two radii (like a slice of pie). A segment is the part of a disk between an arc and the chord joining its ends.
Formulas (angle α in degrees)
| Value | Formula |
|---|---|
| Arc length | L = πrα / 180° |
| Sector area | S = πr²α / 360° = L · r / 2 |
| Chord | c = 2r · sin(α/2) |
| Segment height | h = r(1 − cos(α/2)) |
| Segment area | Sc = S − ½r² sin α |
| Sector perimeter | P = 2r + L |
With the angle in radians the formulas are simpler: L = rα, S = r²α / 2.
Example. r = 6 cm, α = 60°.
L = π · 6 · 60 / 180 = 2π ≈ 6.28 cm
S = π · 36 · 60 / 360 = 6π ≈ 18.85 cm²
c = 2 · 6 · sin 30° = 6 cm
Sc = 6π − ½ · 36 · sin 60° ≈ 18.85 − 15.59 = 3.26 cm²
For an angle greater than 180° the segment formula gives the area of the larger segment — the triangle area is added to the sector area.
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