Arc Length & Chord Calculator: arc, chord, sector, segment and sagitta
Find a circle arc’s length, chord, sector area, segment area and height (sagitta) from any two of radius, central angle, chord, segment height or arc length, in degrees or radians, with steps.
How it works
θ in radians = degrees × π/180 · Arc s = r·θ · Chord c = 2r·sin(θ/2) · Sector = ½·r²·θ · Segment = ½·r²·(θ − sin θ) · Height h = r·(1 − cos(θ/2)) · From chord and height: r = (c²/4 + h²) ÷ 2h · From chord and angle: r = c ÷ (2·sin(θ/2)) · From radius and chord: θ = 2·asin(c/2r)
An arc is part of a circle’s edge. Its length is the radius times the central angle in radians, so the calculator first turns degrees (or grads) into radians. The chord is the straight line joining the two ends of the arc; dropping a line from the centre to the middle of the chord splits the picture into two right triangles with angle θ/2, which gives c = 2r·sin(θ/2). The sector is the pizza slice bounded by two radii and the arc, and the segment is the thinner piece between the chord and the arc: the sector minus the triangle. The segment height, or sagitta, is how far the arc bulges above the chord. Pick what you know: radius and angle, radius and chord (minor or major arc), chord and height (the usual case when you measure a curved window, arch or bowed board), angle and chord, or radius and arc length. Every length must be in the same unit; areas come out in that unit squared. The table shows the same circle at common angles for comparison.
Worked example
Radius 10 and a 60° central angle: θ = 60 × π/180 = 1.047198 rad. Arc = 10 × 1.047198 = 10.472 units. Chord = 2 × 10 × sin 30° = 10 units. Sector = ½ × 100 × 1.047198 = 52.3599 sq units. Segment = ½ × 100 × (1.047198 − 0.866025) = 9.0586 sq units. Height = 10 × (1 − cos 30°) = 1.3397 units. The arc is 16.67% of the 62.8319-unit circumference.
Sources: Circular segment — Wikipedia · Arc length formulas — Omni Calculator · Arc length calculator — Inch Calculator · Area of a circular segment — Math Open Reference
Frequently asked questions
What is the difference between an arc and a chord?
The arc is the curved part of the circle between two points; the chord is the straight segment joining the same two points. The chord is always shorter, and the gap grows as the angle widens: at 60° the arc is about 4.7% longer, at 180° it is about 57% longer.
How do I find the radius of an arch or curve I can measure?
Measure the chord (the straight span across the ends) and the height of the curve at its middle, then choose “Chord and segment height”. The radius is (c²/4 + h²) ÷ 2h. A 48 in span that rises 6 in has a radius of (576 + 36) ÷ 12 = 51 in.
Why does the angle have to be in radians?
A radian is defined so that an arc of one radius length subtends one radian, which makes s = r·θ and sector = ½·r²·θ exact. In degrees you would need the extra factor π/180, which the calculator applies for you.
How do I find the arc length from the chord and the radius?
First get the angle: θ = 2·asin(c ÷ 2r). Then s = r·θ. A 10 unit chord on a radius-10 circle gives θ = 2·asin(0.5) = 60° (1.0472 rad) and an arc of 10.472. Turn on “major arc” for the long way round, 2πr minus that.
What is the difference between a sector and a segment?
A sector is bounded by two radii and the arc, like a slice of pie. A segment is bounded by the chord and the arc only. Segment area = sector area − triangle area = ½·r²·(θ − sin θ).
What happens at 180° and 360°?
At 180° the chord becomes the diameter, the arc is half the circumference (πr), and the sector and segment are both half the circle. At 360° the arc is the full circumference, the chord shrinks to zero and the sector is the whole circle.
Related calculators
This calculator is for educational purposes. Double-check important results before acting on them.