Circle and Arc Calculator: Chord, Sagitta and Radius from Offset
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Circle and Arc Calculator: Chord, Sagitta and Setting Out
Arc length, chord, sagitta and segment area from a radius and angle, or the radius recovered from a chord and its offset, which is how you find the radius of an existing curve with nothing but a tape.
The field method: stretch a tape between two points on the curve, that is the chord c. Measure from the middle of the tape square to the curve, that is the sagitta or offset h. The radius is c squared over 8h, plus h over 2. A 20 m chord with a 0.5 m offset means a 100.25 m radius.
How to use this tool
Three ways in, one full table out. Give it a radius and central angle, and it returns the arc, chord, sagitta, sector area and segment area. Give it a chord and its mid offset, and it recovers the radius, which is the field problem. Give it a radius and chord, and it finds the angle.
Whichever mode you start in, the full table appears, because once any two compatible quantities are known the whole geometry is fixed.
The sagitta, and why a tape can find a radius
Stretch a string between two points on a curve and you have a chord. The little gap between the middle of that string and the curve itself is the sagitta, from the Latin for arrow, the bow and bowstring picture being exactly right.
The useful fact is that the radius is recoverable from just those two tape measurements: r equals c squared over 8h, plus h over 2. No instrument, no coordinates, no access to the centre, which is usually somewhere in the traffic anyway. This is how you find the radius of an existing kerb line, a rail curve or a formed arch with nothing but a tape and a straight edge.
It is worth seeing why it works: the sagitta, the half chord and the radius sit in a right triangle with hypotenuse r and legs r minus h and c over 2. Pythagoras, expanded and rearranged, gives the formula. Everything in this tool is that one triangle wearing different clothes.
A worked check you can repeat on paper
A kerb line: 20 m chord, and the offset from the string to the kerb at mid span tapes at 0.5 m. Radius is 400 over 4, plus 0.25, which is 100.25 m. If the drawing says the kerb radius is 100, your as built is a quarter metre generous, and now you know without setting up an instrument.
The reverse check: a radius of 10 with a 60 degree angle gives a chord of exactly 10, because the chord and two radii form an equilateral triangle at 60 degrees. Feed that chord and its sagitta back through the field formula and the 10 comes back exactly, which is how the implementation was verified.
Mistakes that catch people out
Measuring the offset somewhere other than mid chord. The formula wants the sagitta at the exact middle. Fix: mark the midpoint of the string before measuring.
Confusing arc length with chord length. The arc is always longer, and at small angles they are deceptively close, which hides errors until they matter. Fix: at 60 degrees the arc is already 4.7 percent longer; never substitute one for the other.
Mixing degrees and radians in the arc formula. Arc equals r theta only in radians. Fix: this tool takes degrees and converts internally; hand calculations need the pi over 180.
An offset bigger than half the chord. That would put the point beyond a semicircle, which a mid chord offset cannot do. Fix: the tool rejects it; the usual cause is measuring to the wrong reference.
How this was checked
The equilateral case returns a chord of exactly the radius at 60 degrees. The semicircular segment returns exactly half the disc area. The setting out case of a 20 m chord and 0.5 m offset returns exactly 100.25 m, and every radius and angle pair survives the round trip out to chord and sagitta and back through the field formula. A chord equal to the diameter returns exactly 180 degrees.
Related Calculators and Reference
Sector areas connect to the Area of 2D Shapes Calculator, and the right triangle at the heart of the sagitta formula is the subject of the Distance and Pythagoras Calculator. Central angles in gradians, which European setting out instruments use, convert in the Angle Converter. For borehole rather than road curvature, the Dogleg Severity Calculator speaks the same geometry in degrees per 30 metres.