Two planes and the gap between them
All of this is one piece of geometry. Lay a straightedge along the tops of the frets and let it run past the end of the fingerboard to the bridge. It is now describing the fret plane. The soundboard is the other plane. The angle between them is what this page calls the neck angle, and the gap between them at the bridge is what decides how much saddle there is.
Two numbers pin the fret plane down. One is where it sits at the body joint, which you can measure directly: the height from the soundboard to the top of the fret over the joint, which on a flat-top build is the fingerboard thickness plus the fret height. The other is the angle. Given both, the height the plane reaches at the bridge is the joint height plus the reach from joint to bridge times the tangent of the angle.
On a 25.5 inch scale with the saddle at 25.6 inches and the joint at the fourteenth fret, that reach is 11.459 inches. At three tenths of a degree the plane climbs 60 thousandths over it. At one degree it climbs 200 thousandths. That single figure, two tenths of an inch per degree, is the one worth carrying around.
Where the string sits above the plane
The saddle is not on the fret plane, it is above it by whatever the string height at the bridge works out to. That comes straight off the action. The string runs from the nut, effectively touching the fret plane there, to the top of the saddle, so its height above the plane grows in proportion to the distance from the nut. At the twelfth fret, which is half the scale, it has climbed roughly half the way.
Roughly, not exactly, because the saddle is at the scale length plus compensation rather than at the scale length. With the saddle at 25.6 inches and the twelfth fret at 12.75, the ratio is 2.008 rather than 2. It is a small correction and it is free to include, so the page includes it.
So 90 thousandths of action at the twelfth fret puts the string 181 thousandths above the fret plane at the saddle. Add the plane height at the bridge and you have the saddle top above the soundboard. Subtract the bridge height and you have what stands proud.
Why a tenth of a degree matters and a tenth of an inch of anything else does not
The angle is multiplied by the reach, and everything else is not. Twenty thousandths of extra fingerboard thickness moves the plane at the bridge by twenty thousandths. A tenth of a degree of angle moves it by twenty thousandths too, and a tenth of a degree is a slope of under two thousandths of an inch per inch, which is invisible.
That asymmetry is the whole reason neck angle gets cut on a jig, checked with a straightedge over the bridge, and adjusted in fractions nobody can see. It is also why an instrument that has settled over decades, with the top bellying up a little and the neck rotating a little, ends up with no saddle left. Neither movement was large. Both were multiplied by eleven inches.
Two ways round the same equation
The page runs the geometry in both directions because builders and repairers approach it from opposite ends. A builder knows the saddle height they want to end up with, usually because they want enough material to adjust later and enough break angle over the top, and needs the angle that produces it. A repairer has an angle in front of them, whatever the instrument has, and needs to know what it leaves.
The angle it reports for a wanted saddle is the honest answer to that geometry and not a recommendation. What saddle height a design wants is a decision about that design, involving the bridge, the break angle, the pickup if there is one, and how the instrument is meant to be maintained.
What this leaves out
Relief, for one. The fret plane is not flat on a strung instrument, and the straightedge is describing the frets as levelled rather than the frets as loaded. On an assembled instrument the projection is normally taken with the neck under tension, which folds the bow in whether you meant it to or not.
Nut height, for another. The model has the string touching the fret plane at the nut, which it never quite does. The error is small at the bridge because it enters through the proportion of the way along, and the nut is at zero. Top movement is the third: a soundboard under string load rises behind the bridge and sinks in front of it, so the plane the bridge sits on is not the plane it was glued to.
Where this connects
The action figure this page consumes is set on the string action and saddle height calculator, which is also where the two to one lever comes from. The bow left out of the fret plane is the neck relief calculator, and the other end of the string is the nut slot depth calculator. The load doing all this over time is the string set pulling on the neck and the top, which the string tension calculator totals. For the angle itself as a slope rather than a degree, the angle and slope calculator converts between the two.
Questions people ask
What exactly is the neck angle measured against?
On this page, the plane of the fret tops against the plane of the soundboard, with a positive angle meaning the fret plane rises as it goes toward the bridge. That is the definition that makes the arithmetic work, because it is the one that tells you where a straightedge on the frets lands at the bridge. Plans and shop drawings use other references — the body sides, the centreline, the plane of the neck blank before the fingerboard goes on — and the numbers from those are not interchangeable with this one. Always check what a quoted angle is measured against before feeding it in anywhere.
Why is a tenth of a degree worth caring about?
Because it is multiplied by the distance from the joint to the bridge, which on a common flat-top layout is about eleven and a half inches. One degree over that reach is two tenths of an inch. A tenth of a degree is twenty thousandths, which is a noticeable fraction of the saddle standing above a bridge. Nothing else in the geometry gets multiplied like that: a twenty-thou error in fingerboard thickness moves the bridge end by twenty thou, while a twenty-thou movement at the bridge from the angle takes a slope you cannot see by eye.
Does the saddle height come out of this or does the action?
Both, depending on which one you hold fixed, and the page runs it either way. If you fix the action you want, the angle decides how much saddle stands above the bridge. If you fix the saddle, the angle decides the action. Tilting the neck back a degree buys about two tenths of an inch more saddle at the same action, or takes about a tenth of an inch out of the action at the twelfth fret at the same saddle. They are the same fact stated from two ends, which is why the page reports both figures.
What does it mean when the saddle has run out?
That the geometry cannot deliver the action being asked for, and no more can be taken off something that is already flush with the bridge. On an instrument that has been played for decades this usually accumulates from two small movements: the top rising slightly behind the bridge under string load, and the neck rotating slightly forward. Neither is large on its own; both get multiplied by the reach to the bridge. What that situation calls for is structural work on the neck joint, which is a different kind of job from a setup, and it belongs to whoever does that kind of work.
Should the straightedge land on top of the bridge or somewhere else?
That is a design decision and this page deliberately does not answer it. Where a straightedge lands depends on the bridge height, the saddle height the design intends, and how much of a break angle the strings need over the top of the saddle, and different instruments are built to land in different places. What the page does is tell you where a given angle puts it, and what a given landing point implies about the angle, so that whatever target you or your plan have chosen can be worked to rather than guessed at.