Radius from Chord and Sagitta
This finds the radius of a large circle from a fragment, without locating the centre. Lay a straight edge across the chord and measure the height at the fullest point. It is how you recover the curvature of a broken plate or an arch.
Your numbers
Radius
160.25cm
Diameter
320.5cm
Angle (°)
36.36°
Arc length
101.698cm
Formula
Radius = (Chord length ² ÷ 4 + Sagitta ²) ÷ (2 × Sagitta)
A very small sagitta makes the radius blow up and amplifies measurement error with it. Use a longer chord for accuracy.
What to enter
| Input | Default | Accepted range |
|---|---|---|
| Chord length (cm)The straight line between two points on a circle. | 100 | 0 and up |
| Sagitta (cm)The height from the middle of the chord to the arc — how much it bulges. | 8 | 0.01 and up |
Step by step
Quick reference table
Results when only Chord length (cm) changes and everything else stays put.
| Chord length (cm) | Radius (cm) | Diameter (cm) | Angle (°) (°) |
|---|---|---|---|
| 50 | 43.063 | 86.125 | 70.98 |
| 75 | 91.891 | 183.781 | 48.17 |
| 100 | 160.25 | 320.5 | 36.36 |
| 150 | 355.563 | 711.125 | 24.35 |
| 200 | 629 | 1,258 | 18.3 |
What each result means
| Result | At default values |
|---|---|
| Radius (cm)Centre to edge — half the diameter. | 160.25 |
| Diameter (cm)The longest line across a circle — twice the radius. | 320.5 |
| Angle (°) (°)An angle in degrees; a full turn is 360°. | 36.36 |
| Arc length (cm)The length of the curved piece cut from a circle’s edge. | 101.698 |
Common mistakes
A very small sagitta makes the radius blow up and amplifies measurement error with it. Use a longer chord for accuracy.
Glossary
- Chord length
- The straight line between two points on a circle.
- Sagitta
- The height from the middle of the chord to the arc — how much it bulges.
- Radius
- Centre to edge — half the diameter.
- Diameter
- The longest line across a circle — twice the radius.
- Angle (°)
- An angle in degrees; a full turn is 360°.
- Arc length
- The length of the curved piece cut from a circle’s edge.
Frequently asked questions
Q. How is Radius from Chord and Sagitta calculated?
Radius = (Chord length ² ÷ 4 + Sagitta ²) ÷ (2 × Sagitta) — This finds the radius of a large circle from a fragment, without locating the centre. Lay a straight edge across the chord and measure the height at the fullest point. It is how you recover the curvature of a broken plate or an arch.
Q. Can you walk through an example?
With Chord length 100cm, Sagitta 8cm, the answer is Radius 160.25cm.
Q. What do I need to enter?
Enter Chord length, Sagitta. The result recalculates as you type, and an empty box counts as zero.
Q. How much does the answer move if I change a number?
Changing only Chord length (cm) moves the answer to Chord length (cm) 50 → Radius (cm) 43.063 and Chord length (cm) 200 → Radius (cm) 629. The table below lays out five steps.
Q. How are the numbers rounded?
Money is shown to the nearest whole unit, percentages to one decimal place and everything else to two. What you see is rounded; the calculation itself carries the unrounded value forward.
Q. Anything to watch out for?
A very small sagitta makes the radius blow up and amplifies measurement error with it. Use a longer chord for accuracy.
Related calculators
Keep your length units consistent across inputs — mixing centimetres and metres throws results far off.