Math Tables
Circle Chord, Sagitta and Arc Table
Three lengths describe the same piece of a circle, and they are easy to confuse because they are all "the width of the curve". The chord is the straight line between two points on the circle. The arc is the curved path between those same points. The sagitta — also called the rise or the height of the segment — is the distance from the midpoint of the chord up to the arc. Every layout problem involving a curved edge needs two of the three, and the tables here convert between them.
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All three follow from the central angle and the radius, and the two formulas worth memorising are chord = 2R·sin(θ/2) and sagitta = R(1 − cos(θ/2)). The arc is simply R·θ with θ in radians, which is the definition of the radian. The reason a table is useful is that these are trigonometric, not linear: doubling the angle does not double the chord. A 30° chord is 0.518 of the radius while a 60° chord is 1.000 — nearly double for exactly double the angle, but a 120° chord is 1.732, not 2.
The reverse direction is the one that gets used on site. You can measure a chord with a tape and a sagitta with a straightedge and a rule, but you cannot measure a radius directly if the centre is inside a wall or above a ceiling. The formula R = c²/(8s) + s/2 recovers the radius from those two measurements, and it is exact for a circular arc — no iteration, no approximation beyond the measurement itself.
Chord, sagitta and arc for a unit radius, by central angle
| Angle (°) | Chord | Sagitta | Arc | Arc as share of full circle |
|---|---|---|---|---|
| 0 | 0.000000 | 0.000000 | 0.000000 | 0.00% |
| 10 | 0.174311 | 0.003805 | 0.174533 | 2.78% |
| 20 | 0.347296 | 0.015192 | 0.349066 | 5.56% |
| 30 | 0.517638 | 0.034074 | 0.523599 | 8.33% |
| 45 | 0.765367 | 0.076120 | 0.785398 | 12.50% |
| 60 | 1.000000 | 0.133975 | 1.047198 | 16.67% |
| 75 | 1.217523 | 0.206647 | 1.308997 | 20.83% |
| 90 | 1.414214 | 0.292893 | 1.570796 | 25.00% |
| 105 | 1.586707 | 0.391239 | 1.832596 | 29.17% |
| 120 | 1.732051 | 0.500000 | 2.094395 | 33.33% |
| 135 | 1.847759 | 0.617317 | 2.356194 | 37.50% |
| 150 | 1.931852 | 0.741181 | 2.617994 | 41.67% |
| 165 | 1.982890 | 0.869474 | 2.879793 | 45.83% |
| 180 | 2.000000 | 1.000000 | 3.141593 | 50.00% |
At 60° the chord equals the radius — the property that makes a compass setting of one radius divide a circle into six. At 180° the chord is the diameter (2) and the sagitta is the radius (1), which is the sanity check for both formulas.
Chord minus arc: how much shorter the straight line is
| Angle (°) | Arc (R = 1) | Chord (R = 1) | Arc − chord | Difference |
|---|---|---|---|---|
| 30 | 0.523599 | 0.517638 | 0.005961 | 1.14% |
| 45 | 0.785398 | 0.765367 | 0.020031 | 2.55% |
| 60 | 1.047198 | 1.000000 | 0.047198 | 4.51% |
| 90 | 1.570796 | 1.414214 | 0.156583 | 9.97% |
| 120 | 2.094395 | 1.732051 | 0.362344 | 17.30% |
| 150 | 2.617994 | 1.931852 | 0.686142 | 26.21% |
| 180 | 3.141593 | 2.000000 | 1.141593 | 36.34% |
This is the number that matters when you are cutting material for a curved edge: a 90° arc is 10% longer than its chord, while a 30° arc is only 1.1% longer. Small angles let you treat the arc as a straight line without much error.
Radius from a measured chord and sagitta
| Chord (mm) | Sagitta (mm) | Radius (mm) | Central angle (°) |
|---|---|---|---|
| 1,000 | 100 | 1,300 | 45.24 |
| 1,000 | 200 | 725 | 87.21 |
| 1,000 | 300 | 566.7 | 123.86 |
| 500 | 50 | 650 | 45.24 |
| 500 | 100 | 362.5 | 87.21 |
| 2,000 | 250 | 2,125 | 56.14 |
| 2,000 | 500 | 1,250 | 106.26 |
| 3,000 | 300 | 3,900 | 45.24 |
Measure the straight distance across the opening (the chord) and the height at its centre (the sagitta), then read the radius. A 1000 mm chord with a 100 mm rise gives a 1300 mm radius — the formula is R = c²/(8s) + s/2, and the angle column follows from the same triangle.
How to Use These Tables
- For a unit-radius table, scale by the real radius: multiply every length by R. A 90° arc on a 2 m radius is 1.570796 × 2 = 3.1416 m, and its chord is 1.414214 × 2 = 2.8284 m.
- To lay out an arch or a curved counter, measure the chord (the opening) and decide the rise, then use the third table to get the radius. Set a trammel or a string to that radius from the computed centre.
- To estimate material for a curved edge, use the arc-minus-chord table: the arc is 10% longer than the chord at 90°, but only 1.1% longer at 30°. For shallow curves you can cut to the chord and sand the difference away.
- To divide a circle into equal parts with a compass, remember that a 60° chord equals the radius — six steps around the circumference closes exactly. That is why 60° appears in every construction diagram and 90° does not.
Common Questions
What is the difference between sagitta and arc height?
They are the same thing. Sagitta, rise, arc height and segment height all name the perpendicular distance from the chord's midpoint to the arc. The word sagitta is Latin for arrow, from the shape of a bow and arrow.
Can I find the radius if I only know the chord?
No — a chord of a given length fits infinitely many circles, from a nearly flat arc with a huge radius to a semicircle whose radius is half the chord. You need a second measurement, either the sagitta or the central angle, to pin the circle down.
Why is the chord at 60° exactly equal to the radius?
Because a 60° central angle makes an equilateral triangle with the two radii: all three sides are equal, and the third side is the chord. It is the one case where the trigonometry collapses into something you can check by eye, and it is the basis of the classic six-part circle division.
Is the arc always longer than the chord?
Always, for any angle greater than zero, because a straight line is the shortest path between two points. The difference grows with the angle: 1.1% at 30°, 10% at 90°, and 36% at 180° where the arc is π and the chord is 2.
How accurate is R = c²/(8s) + s/2?
It is exact for a true circular arc, so the only error comes from measurement. In practice the sagitta is the weak point: a 1 mm error in a 10 mm rise is a 10% error, while the same 1 mm in a 1000 mm chord is 0.1%. Measure the rise carefully and keep the chord as long as practical.
More Math Tables
Sources
- Circular segment — chord, sagitta and arc relations — CRC Standard Mathematical Tables and Formulae
- Circular segment — formulas and derivations — Wolfram MathWorld
- NIST Handbook 44, Appendix C — length measurement tolerances — National Institute of Standards and Technology
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