Formula used
- R = 1 / |κ| when curvature κ is known.
- R = v² / a for circular motion with centripetal acceleration a.
- R = v / ω using speed v and angular speed ω.
- R = s / |θ| from arc length s and turning angle θ in radians.
- R = c²/(8h) + h/2 from chord c and sagitta h.
How to use this calculator
- Select the method that matches your known quantities.
- Enter values and pick the correct units for each field.
- Choose the output unit for the radius value.
- Press Calculate to see results above the form.
- Use the download buttons to export your computed table.
Example data table
| Method | Inputs | Radius R (m) | Curvature κ (1/m) |
|---|---|---|---|
| From κ | κ = 0.25 1/m | 4.0000 | 0.2500 |
| From v and a | v = 12 m/s, a = 3 m/s² | 48.0000 | 0.020833 |
| From v and ω | v = 9 m/s, ω = 1.5 rad/s | 6.0000 | 0.16667 |
| Chord and sagitta | c = 0.30 m, h = 0.015 m | 0.76875 | 1.30081 |
Technical article
1) Why radius of curvature matters
Radius of curvature, R, turns “how bent” into a measurable length. In vehicle dynamics, R controls lateral acceleration through a = v²/R, helping set safe cornering speeds. In optics and metrology, R characterizes curved surfaces, such as mirrors and lenses, where small changes in R alter focus and alignment.
2) Curvature versus radius
Curvature is the inverse scale: κ = 1/R (units of 1/length). Many datasets store κ because it varies smoothly along a path, while R can blow up when a curve becomes nearly straight. This calculator reports both values so you can switch between geometric intuition (R) and differential description (κ).
3) Speed and centripetal acceleration
If a moving object follows a circular arc, centripetal acceleration satisfies a = v²/R. For example, at v = 20 m/s on a 100 m curve, a ≈ 4.0 m/s² (about 0.41 g). Rearranging yields R = v²/a, which is useful when acceleration comes from sensors or force measurements.
4) Speed and angular speed
When angular speed is known, v = ωR gives R = v/ω. This route is common in rotating rigs and turntable tests. Converting degrees per second to radians per second is essential: 60 deg/s equals about 1.047 rad/s. The calculator handles both options and displays converted SI values for traceability.
5) Arc length and turning angle
For circular motion, s = Rθ with θ in radians. If a robot travels s = 5 m while turning θ = 10° (0.1745 rad), then R ≈ 28.65 m. This method is robust for gentle turns, but it becomes sensitive when θ is very small because small heading errors produce large changes in R.
6) Chord and sagitta measurements
Surface profiling often measures a chord length c and sagitta h. The circle relation R = c²/(8h) + h/2 avoids direct angle measurement. As an example, c = 300 mm and h = 15 mm gives R ≈ 768.75 mm. Small h values can amplify measurement noise, so record sufficient precision.
7) Units, scaling, and reporting
R is a length, so unit mistakes can shift results by orders of magnitude. A value entered in centimeters must be converted to meters before combining with SI accelerations. This calculator supports common length, speed, and acceleration units (including g) and exports a summary table so your report preserves assumptions, method, and output units.
8) Quality checks and interpretation
Sanity checks prevent misinterpretation. For chord–sagitta geometry, R should exceed h and typically exceed c/2 for small segments. For motion methods, compare computed κ = 1/R across different measurements; agreement suggests consistent data. If results differ, revisit unit choices, sensor calibration, and whether the path truly approximates a circular arc.
FAQs
1) Is the radius always positive?
The radius magnitude is positive. Direction is usually captured by the sign of curvature or turning angle. This calculator reports a positive radius and also shows κ so you can apply your sign convention externally.
2) What if κ is negative?
Negative curvature typically indicates the curve bends in the opposite direction. The magnitude still gives the same radius size. Enter κ as measured; the calculator uses |κ| for R and reports κ = 1/R as a positive magnitude.
3) Why do some results look extremely large?
Very small curvature, small angles, or very low centripetal acceleration imply a nearly straight path, producing a large R. This is physically reasonable, but it also means the estimate can be sensitive to noise.
4) When should I use R = v/ω instead of R = v²/a?
Use v/ω when ω is directly measured and stable, such as in a rotating rig. Use v²/a when acceleration is measured reliably and motion is close to uniform circular motion.
5) Does the chord–sagitta formula assume a perfect circle?
Yes, it assumes the measured segment is part of a circle. If the surface is aspheric or the path curvature changes rapidly, the computed R is a local approximation over that chord.
6) Can I export multiple runs into one file?
The built-in export saves the current result. For multiple runs, compute each case and save the CSV files, then combine them in a spreadsheet or script for batch reporting.
7) How accurate is the g conversion?
The calculator uses standard gravity g = 9.80665 m/s². For high-precision work, confirm whether your instrument defines g locally or uses a slightly different calibration factor.