Where the relation comes from
Take a cross-section straight through the blade, perpendicular to the edge. At the top of the bevel the steel is as thick as the stock. At the edge it is as thick as whatever you left. Between those two points the ground face is a straight line, and the bevel height is how far apart they are measured up the flat of the blade.
On a grind that is the same both sides, each face has to remove half the difference between those two thicknesses. So one face steps away from the centreline of the blade by half of (stock minus edge), over a distance equal to the bevel height. The tangent of the angle that face makes with the centreline is therefore half the thickness difference divided by the bevel height, and the two faces together make twice that.
On a chisel grind only one face moves, so it removes the whole difference rather than half of it, and the included angle is the single angle between that face and the flat back. Same relation, one instead of two.
Note what is not in that anywhere: any absolute dimension. The angle is set by a ratio, so a blade twice the size with everything doubled has exactly the same grind. That is why the angle is a consequence rather than a setting.
Measured up the flat, not along the face
There are two ways to measure a bevel height and they differ. Scribe a line on flat stock before you grind and you are measuring the projection — the distance up the blade in the plane of the flat. Measure the finished ground face with a rule laid on it and you get the slant, which is the hypotenuse of that projection and the thickness removed.
This page uses the projection, because that is the line people scribe. At the angles a knife bevel actually runs, the two differ by a few thousandths: on a 0.9 inch bevel removing 0.068 inch per side, the slant is longer by about 0.0026 inch. That is inside the width of a scribe line, so the distinction rarely matters — but it grows fast on steep bevels, and the page prints both so you can see when it starts to.
The primary grind is not the edge
| Primary bevel | Sharpened edge | |
|---|---|---|
| Runs over | The whole bevel height, often an inch or more | The last few thousandths |
| Set by | Stock, edge thickness and bevel height | Whatever you sharpen at |
| Typical magnitude | Single figures in degrees | Several times steeper |
People conflate these constantly, and it is the source of most of the confusion around blade angles. The number this page produces is the geometry of the grind — what governs how the blade goes through material behind the edge. The angle you sharpen at is a separate decision made on a separate, much smaller facet, and it is what governs how the very edge behaves. A blade can have a thin primary grind and a stout sharpened edge, or the reverse, and those two blades behave nothing alike.
Why nobody can tell you the right number
Because it is a trade rather than a fact. Thinner geometry parts material with less force and gives up edge stability doing it; thicker geometry survives more and pushes harder. Where the sensible compromise sits depends on the steel, on how it was treated, on what the blade is going to be asked to do, and on how much the owner minds putting an edge back on it.
Those things cannot be resolved by arithmetic, and any page that hands you a single correct angle is guessing on your behalf. What arithmetic is good for is the other direction: you decide the geometry, and it tells you exactly what you are about to make, and exactly how much that changes if the next bar comes in twenty thousandths thicker.
The legal side is worth naming separately and it is not a matter of geometry at all. What may be made, owned, carried, sold or shipped is set by state and local law, it varies enormously and it changes. Those questions go to a lawyer and to the relevant authority, and nothing on this page addresses them.
Questions people ask
What included angle should I grind my knife at?
This page will not tell you, and neither should anything else without knowing a great deal more. The sensible geometry depends on the steel, on how it was heat treated, on what the blade is for and on how much resharpening the owner will tolerate, and those pull against each other. What the page does instead is the honest half: you choose the geometry and it tells you exactly what you are producing, including how much the answer moves when the stock thickness changes.
Why did the same grind come out at a different angle on new stock?
Because the angle is a consequence of the stock thickness, not a setting. Holding the bevel height and the edge thickness the same, ten thousandths of extra stock opens the included angle measurably — the page prints the exact figure for your dimensions. Bar sold as one thickness routinely arrives over or under, which is why measuring the actual bar with a caliper before you scribe the bevel line is worth the thirty seconds.
Is the bevel height measured up the flat or along the ground face?
Up the flat, which is the projection you scribe on the blank before grinding. The ground face is the hypotenuse of that height and the thickness removed, so it is slightly longer — on a 0.9 inch bevel removing about 0.068 inch per side, roughly 2.6 thousandths longer. The page prints both. At knife bevel angles the difference sits inside a scribe line, but it grows quickly on steep bevels.
How is a chisel grind different in the arithmetic?
One face does all the work instead of two sharing it. On a symmetric grind each side removes half the difference between the stock and the edge thickness; on a chisel grind one side removes all of it over the same bevel height, so the face sits at roughly twice the angle. The included angle is then measured between that single ground face and the flat back rather than between two ground faces.
Does this give me the angle to set my sharpening jig to?
No. The angle here belongs to the primary grind, which ends at whatever thickness you left at the edge. The facet you sharpen is a separate and much steeper one applied over the final few thousandths, and its angle is a separate decision. What the primary grind governs is how the blade behaves behind the edge; what the sharpened facet governs is how the very edge behaves. They are different numbers for different reasons.