Arc length calculator: arc length, chord, sagitta and radius
Enter two measurements of the arc and get all the others: arc length, chord, sagitta (rise), radius, central angle, tangent and areas, with a drawing to scale.
Use any unit (meters, centimeters, feet…): the result comes out in the same unit. The angle accepts decimal degrees (60.5) or degrees, minutes and seconds (60°30′).
Result
Enter the two measurements above.
Arc formulas
With the radius R and the central angle Δ (in radians = degrees × π ÷ 180): arc L = R × Δ · chord C = 2R × sin(Δ/2) · sagitta s = R × (1 − cos(Δ/2)) · tangent T = R × tan(Δ/2).
Radius from chord and sagitta: R = (C² ÷ 4 + s²) ÷ (2s). It is the easiest way to find the radius of an existing arc: stretch a string between the ends (chord) and measure the largest distance to the arc (sagitta).
Areas: sector = R² × Δ ÷ 2; segment (between the arc and the chord) = R² × (Δ − sin Δ) ÷ 2.
FAQ
How do you calculate the length of an arc?
Multiply the radius by the central angle in radians: L = R × Δ. With the angle in degrees, L = 2πR × Δ ÷ 360. E.g.: radius 50 m and angle 60° → L = 50 × 1.0472 = 52.36 m.
How do you find the radius of an arc from the chord and the sagitta?
Use R = (C² ÷ 4 + s²) ÷ (2s). E.g.: chord 8 m and sagitta 1 m → R = (16 + 1) ÷ 2 = 8.5 m.
What is the difference between an arc and a chord?
The arc is the curved path along the circle; the chord is the straight line joining its two ends. The chord is always shorter than the arc, and the difference grows with the angle.
What is the sagitta of an arc?
It is the largest distance between the arc and the chord, measured at the middle, perpendicular to the chord. In road design and surveying it is also called the middle ordinate.