Circular Arc Sagitta Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The sagitta of a circular arc — the distance between the midpoint of the chord and the highest point of the arc — is calculated with s = r − √(r² − (L/2)²), where r is the radius and L the chord length. For a circle of radius 5 cm with a 5 cm chord, the sagitta is about 0.67 cm.
Explanation
The sagitta measures how much a circular arc "bulges" relative to its chord: it is the distance between the midpoint of the chord and the farthest point of the arc, measured perpendicular to that chord. This calculator directly complements our circle chord length calculator: the two quantities follow from the same geometry (a radius and a central angle, or equivalently a radius and a chord), but answer different questions — one gives a straight-line distance between two points of the circle, the other gives the height of the arc portion separating them. The sagitta is an essential quantity in architecture and engineering: it determines the curvature of a bridge arch, a vault, or an optical lens, and appears directly in calculating the radius of curvature of a road or a railway from field measurements (a classic surveying method, where the exact radius of a curve is hard to measure directly, but the chord and sagitta are easy). If the entered chord exceeds the circle's diameter (L > 2r), the configuration becomes geometrically impossible — no chord can be longer than the diameter, the longest possible chord of a circle — and this calculator returns an empty result rather than a meaningless number.
Example: radius 5 cm, chord of 5 cm
Inputs
Radius: 5 cm. Chord length: 5 cm.
Calculation
s = 5 − √(5² − (5÷2)²) = 5 − √(25 − 6.25) = 5 − √18.75 ≈ 5 − 4.3301 ≈ 0.6699 cm.
Result
The sagitta of this circular arc is about 0.67 cm.
Frequently asked questions
What happens if the chord equals the diameter?
In that limiting case, the chord crosses the circle at its exact center (it is a diameter), and the corresponding arc becomes a perfect half-circle: the sagitta is then exactly the radius (s = r), since the highest point of the arc is directly above the center of the circle, at a distance from the chord equal to the radius.
Why can this calculator return an empty result?
Because no chord of a circle can be longer than its diameter (the longest possible chord): if the entered length exceeds twice the radius, the term under the square root in the formula becomes negative, a geometrically impossible configuration. This calculator then shows an empty result rather than a meaningless number, exactly the same behavior as for an impossible triangle in our other geometry calculators.
What is the sagitta of an arc used for in practice?
It is commonly used in surveying to determine the radius of a curve (a road, a railway, a tunnel) from simple field measurements: the chord and the sagitta are easily measured with a tape and a ruler, whereas the radius itself would often be impossible to measure directly, its center generally being inaccessible or far from the site. The actual arc length is calculated separately from the radius and central angle — see our arc length calculator.