What this calculator does
A curve on site is rarely measured by its radius, because the centre of the circle is usually somewhere inaccessible. What can be measured is a straight chord across the arc and the height from that chord to the curve at its midpoint, which is called the sagitta.
Those two measurements determine the radius exactly. The relationship is used for setting out curved walls and roads, checking the curvature of glass and mirrors, and measuring rail and pipe bends.
The formula
Square the chord, divide by eight times the sagitta, and add half the sagitta. The result is the radius of the circle the arc belongs to. From that, the included angle and the arc length follow from ordinary circle geometry.
| Term | Meaning |
|---|---|
| Chord | The straight line joining the two ends of the arc. |
| Sagitta | The perpendicular height from the middle of the chord up to the arc. The name comes from the Latin for arrow, since the chord and sagitta resemble a drawn bow. |
| Included angle | The angle the arc subtends at the centre of the circle. |
The inputs explained
| Field | What to enter |
|---|---|
| Chord (width across the arc) (mm) | The straight-line distance between the two ends of the arc, in millimetres. |
| Sagitta (height at the middle) (mm) | The height from the middle of that chord to the arc, measured perpendicular to the chord. |
When to use it
Setting out a curved wall
Measuring a chord and its sagitta on an existing curve gives the radius, which is what a setting-out drawing needs.
Checking a bend
For a bent pipe or rail, the chord and sagitta can be measured with a straight edge and a rule, which is more practical than locating the centre of the arc.
Measuring optical curvature
The same geometry gives the radius of curvature of a lens or mirror surface from a measured sag over a known aperture.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the radius change with the sagitta, across a fixed 2 metre chord?
The same chord width with a range of measured heights at the middle.
| Sagitta | Radius of curvature | Included angle |
|---|---|---|
| 50 mm | 10,025.00 mm | 11.45° |
| 100 mm | 5,050.00 mm | 22.84° |
| 150 mm | 3,408.33 mm | 34.12° |
| 200 mm | 2,600.00 mm | 45.24° |
| 300 mm | 1,816.67 mm | 66.80° |
Questions
Why is there a "plus half the sagitta" term?
The simpler form, chord squared over eight times the sagitta, is an approximation that is accurate for shallow arcs. Adding half the sagitta makes it exact for any arc, and the difference only becomes visible when the curve is deep relative to its chord.
What if I measure a longer chord?
You get the same radius, since the radius is a property of the circle rather than of where you measured. A longer chord gives a larger sagitta in proportion, and the formula returns the same answer.
Can this find the curvature of something that is not a circular arc?
It assumes a circular arc. For a curve that is not circular, such as a parabolic or a transition curve, the result is the radius of the circle that best fits the two measurements, which is a local approximation rather than a description of the whole curve.
What is curvature as opposed to radius?
Curvature is the reciprocal of the radius. A large radius means a gentle curve and a small curvature value; a tight bend has a small radius and a large curvature.
For the geometry of a slice of a circle, see the circle sector calculator. For arc length from an angle, see the arc length calculator.