Sagitta, which is the only number that matters
A radius on its own tells you nothing about how a board feels, because it is only half the description. What you feel is how far the middle of the board stands above its edges, and that depends on the width being spanned as much as on the radius. The distance is called the sagitta, from the Latin for arrow, because it is the arrow of the bow the arc makes.
It comes out of Pythagoras in one line: for a radius R and a width W, the rise is R minus the square root of R squared minus W over two, squared. A ten inch radius across a 1.6875 inch nut gives 35.7 thousandths of an inch. That is smaller than most people picture, and it is why radius arguments settled by eye are rarely settled correctly.
Because it is a squared term, doubling the width more than doubles the rise. The same ten inch radius across a 2.1875 inch board would give 60.0 thousandths, not the 46 that scaling the rise with the width would suggest. Widths matter more than radii do.
Compound radius, and the constant-rise trick
A compound board flattens as it goes up the neck, on the argument that a tighter radius suits chords at the nut and a flatter one lets a bent string run out to the twelfth fret without choking on the arc. The numbers usually quoted are something like ten inches at the nut going to sixteen at the body.
What is interesting is what that pair does to the rise. Ten inches across 1.6875 gives 35.7 thousandths, and sixteen inches across 2.1875 gives 37.4. Almost identical. The board gets wider, which would raise the rise, and it gets flatter, which lowers it, and the two roughly cancel. So a compound board is not just a flattening board: it is a board where the height of the arch stays put while its shape changes.
Why a compound board is not a cone
The usual shorthand is that a compound fingerboard is a section of a cone. That is a useful mental model for grinding one, because a cone can be generated by a straight-line sweep, but it is not what the common radius pairs actually describe.
On a true cone, everything scales together from the apex. The radius at any station and the width at any station are both proportional to the distance from the apex, so if you extend both back until they reach zero, they reach zero at the same place. Take the ten-to-sixteen board above: the radius reaches zero 30.6 inches behind the nut, and the width reaches zero 61.9 inches behind it. Those are not the same point, so the surface is not a cone.
Work out what the end radius would have to be for it to be one, and with that taper it comes to about 13 inches, not 16. A true conical board from ten inches at the nut ends much tighter than the pairs people actually build, and its rise would grow toward the body rather than staying flat. So the ordinary compound board flattens faster than a cone on purpose, and the constant rise is the reason. It is a deliberate departure, not a sloppy approximation.
The saddle has a radius too, and it is often a different one
Strings at the bridge sit on saddles set to a radius, and the board carries its own radius forward. If the two do not agree across the string span, the middle strings sit differently over their frets than the outer strings do. The page compares them by working out the rise across the string span for each and taking the difference.
On a board carrying a ten-to-sixteen compound out to the bridge, the projected radius is around eighteen inches at the saddle, which across a 2.0625 inch string span gives 29 thousandths of rise. A twelve inch saddle radius across the same span gives 44. That is fifteen thousandths of difference, which on an instrument whose action lives in the seventy to ninety range is not small.
Whether that is wrong depends entirely on the instrument and the player, and a deliberate mismatch is a normal thing to do. What is not useful is having one without knowing it.
The fretwire number that turns out not to matter
It is tempting to work out the arc length of each fret rather than its straight width, on the reasoning that the wire follows the curve. The arc length is twice the radius times the arcsine of half the width over the radius, and on a fingerboard-sized arc it comes to about two thousandths of an inch more than the straight width. Over twenty-two frets that is under a twentieth of an inch in total.
So the honest answer is that the arc costs nothing in wire, and the figure that actually decides how much you buy is the overhang you cut at each end before nipping it flush. The page computes the arcs anyway, because it is more satisfying to see that a number does not matter than to be told it.
Where this connects
Where the frets sit along the board is the fret position calculator, and the string height over them is the string action and saddle height calculator. The bow along the length, as opposed to the arc across the width, is the neck relief calculator. Where the strings sit across that width is the string spacing layout calculator. For the sanding that turns a flat blank into a radiused one, the sandpaper grit progression calculator covers the sequence, and blank stock comes off the board foot calculator.
Questions people ask
What is sagitta and why is it more useful than the radius?
Sagitta is the height of the arc at its middle above a straight line joining its edges — how far the centre of a radiused board stands proud of its sides. It is more useful than the radius on its own because the radius is only half the description: the same ten inch radius produces 36 thousandths of rise across a 1.6875 inch nut and 60 across a 2.1875 inch board. What a hand feels is the rise, not the radius, which is why boards of the same quoted radius can feel quite different if their widths differ.
Is a compound radius fretboard really a cone?
Not usually, despite the shorthand. A true cone has the radius and the width both growing in proportion to the distance from the apex, so extending each back to zero would reach the same point. For a common ten-to-sixteen compound on a typical taper, the radius reaches zero about 31 inches behind the nut and the width about 62 inches behind it, which are nowhere near each other. Making it a true cone would mean ending nearer 13 inches than 16. Real compound boards flatten faster than a cone, and the effect is to keep the rise across the board nearly constant instead of letting it grow.
Does the saddle radius have to match the fretboard radius?
Not necessarily, and plenty of instruments are deliberately set up with a flatter saddle arrangement than the board. What matters is knowing which you have, because a mismatch shows up as the middle strings sitting differently over their frets from the outer ones. The page works out the rise across the string span for both curves and reports the difference, which on a compound board projected out to a twelve inch saddle can be fifteen thousandths. Whether that is right for a particular instrument is a decision for whoever plays it, not a calculation.
How much extra fretwire does the curve need?
Effectively none. The arc across a fingerboard-sized radius is about two thousandths of an inch longer than the straight width, which over a whole neck adds up to less than a twentieth of an inch. The figure that actually decides how much wire a job takes is the overhang you cut at each end before nipping it flush, which is a working habit rather than a geometric quantity. The page computes the arcs so you can see for yourself how little they add.
Does the rise stay the same all the way up a compound board?
On the common radius pairs it very nearly does, which is the point of them. Ten inches across a 1.6875 inch nut is 35.7 thousandths, and sixteen inches across a 2.1875 inch board is 37.4. The board gets wider, which would raise the arch, and it gets flatter, which would lower it, and the two nearly cancel. Enter your own radii and widths and the table shows what your particular combination does, which is worth checking, because a pair that works on one taper will not stay constant on a different one.