Radius From Chord Length Calculator
Find circle radius from a straight chord and measured rise, or solve the same radius from chord plus angle, arc length, or segment area.
📌Chord-Based Presets
⚙Inputs
Use chord length and sagitta rise when you can measure the height at the midpoint. Formula: R = c² / (8s) + s / 2.
Calculated Circle Geometry
🧪Material and Spec Comparison
📊Reference Tables
| Known Pair | Best Formula | Valid Range | Common Use |
|---|---|---|---|
| Chord + sagitta | R = c² / (8s) + s / 2 | s > 0 and c > 0 | Arch templates, curved shelves, field checks |
| Chord + angle | R = c / (2 sin(θ / 2)) | 0° < θ < 360° | CAD drawings, survey arcs, radial fixtures |
| Chord + arc length | Numerically solves c = 2R sin(L / 2R) | L > c | Flexible trim, bent strips, gasket paths |
| Chord + segment area | A = R²(θ - sin θ) / 2 | A > 0 | Glass, panels, vessel sections |
| Chord to Radius Ratio | Curve Depth | Layout Behavior | Measurement Risk |
|---|---|---|---|
| c/R below 0.25 | Very shallow | Nearly flat curve | Sagitta error greatly changes radius |
| c/R 0.25 to 0.75 | Moderate | Easy shop template | Centerline checks usually reliable |
| c/R 0.75 to 1.50 | Deep segment | Strong visible arc | Check end positions and midpoint twice |
| c/R above 1.50 | Near semicircle | Large rise relative to chord | Confirm minor or major arc intent |
| Layout Material | Typical Thickness | Offset Marking | Spec Note |
|---|---|---|---|
| MDF pattern board | 1/2 in to 3/4 in | Pencil or trammel | Good for router-guided arches |
| Plywood form rib | 3/4 in to 1-1/8 in | Station offsets | Use outer radius for form skin |
| Sheet metal blank | 0.8 mm to 6 mm | Scribe line | Allow for kerf or bend process |
| Acrylic or glass panel | 3 mm to 12 mm | CNC or template | Keep chord endpoints protected |
| Example | Chord | Sagitta | Radius |
|---|---|---|---|
| Cabinet door arch | 24 in | 3 in | 25.5 in |
| Window head | 48 in | 6 in | 51 in |
| Site curb arc | 10 ft | 1 ft | 13 ft |
| Metric panel | 1200 mm | 80 mm | 2290 mm |
💡Layout Tips
Somehow, a little magic happens when you discover a circle whose center point is lost. It could be a rusted out tank panel with the original template long gone. Perhaps it is curved shelf bracket you’d like to copy. It could even be an old arched window you’re gazing through. Whatever the case, the curve exists, but the math of how it’s made is not there anymore.
How can you recover the radius that was lost? With just two easy measures, it seems possible, and you don’t even need a degree in trigonometry to do it. You can do this if you use chord length calculator found on this page: just enter your simple field inputs and get accurate layout information. Easy.
How to Find the Radius of a Lost Circle
It all depends on the connection between an arc and the straight line that’s intersected by that arc. In this analogy, the straight line is the chord, the shortest distance from one end of your curved line to the other. And then the sagitta is height of your curve at half its length. Get those two numbers right and the remaining parts of the circle fall into place.
The calculator up top do the math for you; no need to bang around any square root or squared terms in your brain. You get instant results for the radius, diameter, arc length, and segment area. It’s not about how fast you work; it’s all about precision.
This is where many projects fail: accurately measuring the sagitta. The slight sag in your tape measure and/or lack of truly level straight edge will be magnified greatly when calculating large radii. You can have a very small discrepancy (fraction of an inch) off at the middle and end up changing the final radius by several inches on a wide span. So take a little extra time to check your midpoint measurement twice. It’ll save hours of frustration trying to make the metal or wood fit your marks.
This isn’t the only way it works, though; if you happen to have other information at hand, there is other ways. Perhaps you can measure the actual length of the arc along some flexible trim piece? Or maybe you know the central angle from a CAD drawing. The chord plus arc length mode has no simple algebraic solution, so underneath the hood, the tool do some numerical iteration to figure things out. That takes milliseconds. Then it lets you work with what physical constraints you do have on site or in the shop.
One other practical complication involves material thickness. Your plywood formwork, steel plate, or glass panel has some thickness. Drawing a radius on paper is fine. However, things change when you are cutting something from thick material. In my calculator I include a field for material thickness as the outer edge of the material takes a slightly different track around the curve to the centerline. If you’re doing close tolerance joinery or sheet metal work, don’t ignore the difference between these two. Otherwise, it will result in gaps that no fillers can hide. Before making the initial cut know what radius you’re drawing for, the outer or the inner?
These are quick benchmarks. The tables also help give you quick benchmarks for what to expect in some common situations. For example, a small radius that is quite shallow will be affected greatly by any kind of measurement error. A deeper radius mean the measurement is going to affect it less. Knowing that makes choosing the right way to lay things out ideal to your specific situation. Large radii are better suited to station offsets, because if you try to swing a trammel off the center, that point may be yards from your material and physically impossible to reach.
Remember that all formulas are just a map; physical reality is always the goal. Your measurement is the terrain, while the math gives you the map. As you input your chord and sagitta, consider how these values translate to the real-world material. Do they measure from the outside or the inside? Was the surface flat prior to bending? The fifth decimal place of the answer does not matter as much than these types of context checks.
So what? In the end, this is nothing but applied logic. If you can understand where the depth of an arc comes from (the sagitta) and where it gets anchored (the chord), everything else click. Shapes that were once mysterious become yours again. It’s empowering to know how to calculate the radius using only some lengths, so you can create any arc you wish. You can do this to restore a historic detail or to create your own design. The center point wasn’t realy lost after all; it just waited on the proper data to reveal itself again.
