Chord Arc Length Calculator
Convert chord, radius, sagitta, diameter, angle, or known arc data into centerline arc length, rise, segment area, layout allowance, and equal mark spacing.
Use exact centerline radius for paper, plywood, CAD, and CNC reference arcs.
Thin flexible face material usually needs a light trim allowance at ends.
Flat bar and trim strips are best laid out on the neutral centerline.
Tube bends vary by die radius, wall thickness, and measured centerline path.
Arch openings often need joint tolerance checked at spring line and crown.
CNC arcs should use toolpath centerline and verify chord endpoints after offset.
Use more segments when the chord is long or the arc is shallow and hard to sight.
Keep combined mark and measure error below twice the final fit tolerance.
| Central angle | Arc length factor | Chord factor | Sagitta factor | Segment area factor |
|---|---|---|---|---|
| 15 degrees | 0.2618 x radius | 0.2611 x radius | 0.0086 x radius | 0.0015 x radius2 |
| 30 degrees | 0.5236 x radius | 0.5176 x radius | 0.0341 x radius | 0.0118 x radius2 |
| 45 degrees | 0.7854 x radius | 0.7654 x radius | 0.0761 x radius | 0.0502 x radius2 |
| 60 degrees | 1.0472 x radius | 1.0000 x radius | 0.1340 x radius | 0.0906 x radius2 |
| 90 degrees | 1.5708 x radius | 1.4142 x radius | 0.2929 x radius | 0.2854 x radius2 |
| 120 degrees | 2.0944 x radius | 1.7321 x radius | 0.5000 x radius | 0.6142 x radius2 |
| 180 degrees | 3.1416 x radius | 2.0000 x radius | 1.0000 x radius | 1.5708 x radius2 |
| Known inputs | Radius formula | Angle formula | Arc formula |
|---|---|---|---|
| Radius r and chord c | Use entered r | 2 asin(c / 2r) | s = r theta |
| Radius r and angle theta | Use entered r | Use entered theta | s = r theta |
| Chord c and sagitta h | c2 / 8h + h / 2 | 2 asin(c / 2r) | s = r theta |
| Diameter d and chord c | d / 2 | 2 asin(c / d) | s = r theta |
| Arc s and angle theta | s / theta | Use entered theta | Use entered s |
| Chord c and arc s | Solved iteratively | theta from s/c ratio | Use entered s |
| Application | Best known pair | Reference path | Typical segments | Check value |
|---|---|---|---|---|
| Brick or stone arch | Chord + sagitta | Opening centerline | 8 to 16 | Crown rise |
| Pipe saddle wrap | Diameter + chord | Tube centerline | 12 to 24 | Wrap length |
| Tank shell panel | Radius + angle | Inside or outside wall | 16 to 40 | Panel chord |
| Curved trim strip | Radius + chord | Neutral centerline | 10 to 20 | End angle |
| CNC router template | Radius + angle | Toolpath centerline | As programmed | Endpoint chord |
| Sign channel face | Chord + known arc | Face centerline | 12 to 30 | Trim allowance |
| Arc span | Suggested segments | Mark tolerance | Preferred check | Risk if skipped |
|---|---|---|---|---|
| Under 30 degrees | 4 to 8 | 0.5% of rise | Sagitta | Flat arc error |
| 30 to 90 degrees | 8 to 16 | 0.25% of chord | Midpoint rise | Endpoint drift |
| 90 to 180 degrees | 16 to 32 | 0.2% of arc | Quarter marks | Crowded marks |
| Over 180 degrees | 24 or more | Project specific | Full template | Wrong side of circle |
Sometimes you have to cut a stair nosing for a smooth look, sometimes you have to wrap flat steel around a tank. Until it dawns on you how to do it in your head, it just seems like common sense. And if you don’t get it right, you’ll either run out of material or end up with some kind of joint that won’t close. Knowing how an arc relate to its chord becomes useful knowledge on any shop floor.
After you specify the knowns, the calculator does the math, but knowing what each input represent in physical space will help. Typically, most folks begin with the chord (either the width of the material span or the straight-line distance across an opening). From that, you generally know the arc’s height (called the sagitta) or you know radius of the curve itself. In other words, if you’re laying out a masonry arch, you probably know the doorway width and the height you’d like the crown to be. Whether the measurement was in millimeters or inches doesn’t matter; that chord and rise combo together is unique to that particular circle segment. With both inputs, the tool will solve for whatever’s missing. This allows you to see both the true arc length and central angle at the same time.
Why You Need This Calculator
Why does this matter? Because there’s a difference between an arc and a chord. Curves impose strain, and material have a thickness. When you bend a piece of metal around a corner, the inside edge is compressed and the outside edge stretch. The middle (the neutral axis) retains approximately its initial length. This is also why I included the option to draw a reference path or pick a profile type: so your layout points matches the true length of material you’re required to bend or cut out. To use the example above, if you want to bend a flat piece of metal and choose the outside diameter for your bend radius, then you’ll end up with an arc that’s longer than what your metal can accommodate below it. Most folks make this error; they lay their tape across the visible portion of the curve rather than doing math for the neutral axis. Then they get a result that is just a little too loose or tight.
There’s also a handy list of common angles to use as a sanity check (the reference tables is included on the page). So say I’m working with something that’s sixty degrees, a bit longer than the chord, which is scaled off of the radius. At 90 degrees, the straight line vs. The arc length becomes even more dramatic, about eleven percent difference between the two. That number only gets bigger as the angles increase, which is another reason why a big radius will be carefully marked up instead of just eyeballed. In fact, this tool breaks down those giant arcs into bite-sized pieces that you can get exact marks at so you don’t have to wonder how far around the curve is coming next.
It’s also accurate… if you’re willing to accept a certain margin of error. A sixteenth of an inch might be acceptable for CNC routing or high-end fabrication, but in pipe fitting? That can be the difference between a leak and a watertight seal. That could of been the difference between a leak and a watertight seal. With this calculator, you can specify your tolerance and then tack on a percentage allowance for when you need to trim your pieces down. Nothing in the real world is a nice, clean cut, and there will always be a little bit of waste when you grind or file things down. Adding that into your original cut helps so you don’t end up cutting something and finding out it was a hair too small to do what you wanted with it. Not fun.
Respect the geometry and then commit to the curve From laying out model railroad tracks to wrapping a sign face, the curved path versus the straight one is not only measurable but real. Adding in material thickness plus using the proper radius reference makes it a predictable process rather than a guessing game. The result is parts that fit together because the math was done before touching the metal, which saves both time and money on scrap. Actualy, this tool help you work more comfortabley.
