You can measure the radius of a curve using a simple formula that requires only two measurements: the length of a straight line across the curve (the chord) and the height from that line to the peak of the curve. This chord-and-height method works for everything from a bent pipe to a highway on-ramp, and it needs nothing more than a tape measure and a ruler.
The Chord and Height Method
This is the most practical approach for most situations. Stretch a straight line (a string, a straightedge, or a tape measure) between any two points on the curve. That straight span is your chord, and its length is L. Then measure the distance from the midpoint of that chord straight up (or out) to the curve itself. That perpendicular distance is the height, h, sometimes called the “middle ordinate” or “versine.”
The radius formula is:
r = (L² / 8h) + (h / 2)
For example, say you stretch a string 2 meters across a curved wall and measure 5 centimeters from the string’s midpoint to the wall surface. Convert to consistent units (L = 200 cm, h = 5 cm), and the radius works out to (200² / 40) + (5 / 2) = 1,000 + 2.5 = 1,002.5 cm, or about 10 meters. For gentle curves where h is small relative to L, the second term (h/2) barely matters, so the quick approximation r ≈ L² / 8h gets you close.
The longer your chord, the more accurate your result. A very short chord on a large-radius curve will give you a tiny height measurement, and small errors in that height get magnified in the calculation. When possible, use the longest chord that still fits within the curved section.
Using Three Points on the Curve
If you can identify three points anywhere on the curve, you can find the exact radius geometrically. This works because any three non-collinear points define one and only one circle.
Mark three points (A, B, and C) on the curve. Connect them to form a triangle. Then find the perpendicular bisector of each side of that triangle: measure the midpoint of a side, and draw or snap a line at 90 degrees through it. Where the perpendicular bisectors intersect is the center of the circle. The distance from that center point to any of your three original points is the radius.
On paper or with a CAD program, this is straightforward. In the field, it’s more involved. You’d need to plot your three points on graph paper or enter their coordinates into a calculator. If you know the x,y coordinates of all three points, there are free online circumscribed-circle calculators that will return the radius instantly.
When You Know the Arc Length and Angle
If you can measure how long the curve itself is (the arc length, S) and the angle it sweeps through, the formula is even simpler:
r = S / θ
Here, θ must be in radians. If your angle is in degrees, convert it first by multiplying by π/180, or use the equivalent formula: r = (S × 360) / (2πθ). This method shows up most often in surveying, road design, and CNC machining, where the sweep angle is already specified on a plan or drawing.
The String Line Method for Fieldwork
In railway maintenance and road construction, workers measure curve radius using a variation of the chord method called the string line (or chord versine) technique. A taut wire or string is stretched between two fixed points on the rail or road edge, typically 6, 10, or 20 meters apart depending on the required precision. A ruler or graduated scale is then held at the midpoint of the string, perpendicular to it, to measure the gap between the string and the rail. That gap is the versine, and it feeds directly into the r = L²/8h formula.
This approach is popular because it requires no electronics, works in any weather, and gives repeatable results. Taking versine readings at regular intervals along the curve also reveals whether the curve has a consistent radius or whether it tightens or flattens in spots, which matters for ride quality and safety.
Radius Gauges for Small Curves
For machined parts, fillets, and rounded edges on manufactured objects, a radius gauge set is the fastest option. These are thin metal leaves, each stamped to a specific concave or convex radius. You hold the gauge against the curve and look for light between the gauge and the surface. When no light passes through, you’ve matched the radius.
Professional sets from manufacturers like Mitutoyo include 30 pairs of leaves covering a range of sizes (one common set spans 15.5 mm to 25 mm). Each leaf has both an inside and outside profile, so you can check concave surfaces and convex surfaces with the same tool. For quick shop-floor checks where you need a measurement within a fraction of a millimeter, gauges are faster and more reliable than trying to calculate from chord measurements on a small part.
Digital and Smartphone Tools
Augmented reality apps like AR Ruler use your phone’s camera and motion sensors to measure distances and angles on real-world surfaces. On newer phones equipped with LiDAR sensors, these tools can scan a surface and extract geometric data including curvature. The practical accuracy depends heavily on lighting, surface texture, and phone model. For a rough estimate of a large architectural curve or a landscaping arc, they can save time. For anything requiring precision under a centimeter, stick with physical measurements and the chord formula.
Dedicated surveying instruments, including total stations and laser trackers, can capture hundreds of points along a curve and fit a best-fit circle computationally. These are standard in highway design and industrial fabrication but are expensive and require training.
Choosing the Right Method
- Curved wall, pipe, or archway: Use the chord and height method with a tape measure. It’s quick, cheap, and accurate enough for construction and DIY work.
- Points plotted on a drawing or screen: Use the three-point method or an online circumscribed-circle calculator.
- Road or rail curve in the field: Use the string line versine method with a known chord length.
- Machined part or fillet: Use a radius gauge set for a direct physical comparison.
- Known arc length and sweep angle: Use r = S/θ, converting degrees to radians if needed.
In all cases, the biggest source of error is the height measurement (h) in the chord method, or the perpendicular accuracy in the three-point method. A measurement that’s off by even a millimeter on a small height value can shift your calculated radius significantly. Taking multiple measurements at different points along the curve and averaging the results will give you a more trustworthy number.