Geometry
Arc, sector and circular segment calculator
Enter the radius and just one of angle, arc, chord, sagitta or sector area: every other measurement of the slice follows from it, and the drawing shows the slice to scale.
The parts of a circle
Cut a circle with two radii and you get a sector — the slice of pie — closed off by an arc. Cut it with a chord instead and you get a circular segment: the sector less the triangle standing on that chord. Every quantity involved depends on just two numbers, the radius and the central angle, which is why the radius plus any one measurement is enough here: that one recovers the angle, and the angle gives the rest.
The simplest of the formulas is the arc, provided the angle is in radians: arc = r·θ. In degrees the same thing reads arc = 2πr·θ/360 — the circumference times the fraction of a turn the angle covers — and it is the same reasoning that makes the sector area that fraction of the circle's area. The segment needs one more step: take the sector and subtract the isosceles triangle formed by the two radii and the chord, whose area is ½r²·sin θ.
The sagitta — the height of the segment, from the midpoint of the chord out to the arc — is the measurement that turns up when the curve is a real one rather than a drawing: the radius of a road bend, the rise of a vault, the profile of a masonry arch. It is easy to take on site by stretching a line between two points of the arc and measuring the rise at the middle, and chord and sagitta together give the radius back: r = (chord²/4 + sagitta²)/(2·sagitta).
Common mistakes
- Using arc = r·θ with the angle in degrees. That formula holds in radians only: with θ in degrees it has to be converted first by multiplying by π/180, or the arc comes out about 57 times too long.
- Mixing up the sector and the segment. The sector is bounded by two radii and the arc, the segment by the chord and the arc: the second is always the smaller, and the difference between them is the isosceles triangle between the two radii and the chord.
- Recovering the angle from a chord without asking which arc is meant. The same chord closes both the minor and the major arc, and the arcsine formula always returns the minor one. If the problem is about the larger part of the circle, the angle is 360° minus the one calculated.
Frequently asked questions
How do you find the length of an arc?
With the central angle in radians, arc = r·θ. With the angle in degrees, arc = 2πr·θ/360: you take the circumference and keep the fraction the angle covers. An arc of 90° on a radius of 10 measures 2π·10·90/360 ≈ 15.71.
What is the difference between a sector and a segment?
The sector is the slice bounded by two radii and the arc; the segment is the part between the chord and the arc. The segment comes from the sector by removing the isosceles triangle formed by the two radii: segment area = ½r²(θ − sin θ), with θ in radians.
How do you find the area of a sector?
Area = ½r²θ with θ in radians, or area = πr²·θ/360 with θ in degrees. It is the area of the circle times the fraction of a turn the angle covers. It also equals half the radius times the arc length.
How do you find the radius from a chord and a sagitta?
With r = (chord²/4 + sagitta²)/(2·sagitta). This is how the radius of a curve is measured without reaching its centre: stretch a line between two points on the arc and measure the greatest gap between the line and the arc. You can check it here by entering the radius you found together with the chord.
What is the sagitta of an arc?
It is the height of the circular segment: the distance from the midpoint of the chord out to the arc, measured at right angles to the chord. It is also called the rise. It equals r − r·cos(θ/2), and together with the distance from the centre to the chord it always makes the radius back up.
How this calculation works
With θ in radians: arc = r·θ; chord = 2r·sin(θ/2); sagitta = r(1 − cos(θ/2)); distance from the centre to the chord = r·cos(θ/2); sector area = ½r²θ; segment area = ½r²(θ − sin θ); sector perimeter = arc + 2r; segment perimeter = arc + chord. In degrees: arc = 2πr·θ/360 and sector area = πr²·θ/360. Recovering the angle from the other measurements: from the arc, θ = arc/r; from the chord, sin(θ/2) = chord/(2r), which gives the minor arc, the major one having angle 360° − θ; from the sagitta, cos(θ/2) = 1 − sagitta/r; from the sector area, θ = 2·area/r². Limits: the chord and the sagitta cannot exceed 2r, the arc cannot exceed 2πr, the sector cannot exceed πr². Radius from a chord and a sagitta: r = (chord²/4 + sagitta²)/(2·sagitta).