The two formulas, and the one input everybody gets wrong
Every compound miter reduces to two numbers. Let the half-angle be the amount each of the two pieces must turn to close the corner in plan view, which is 180 divided by the number of sides for a regular figure, or 90 minus half the corner angle for an arbitrary corner. Call that half-angle B. Let S be how far the face of the work leans away from vertical, so S is zero for an upright box side and 90 for a picture frame lying flat. Then, with the stock flat on the saw table:
tan(miter) = sin(S) x tan(B) and sin(bevel) = cos(S) x sin(B)
These are exact. They are not approximations or shop rules of thumb, and they reproduce every published compound angle table if you feed them the right S. Which brings us to the input everybody gets wrong: what the slope is measured from. Crown moulding spring angles are quoted from the wall, meaning from vertical, so 38 degree crown goes in as S equal to 38. Most published compound angle tables for hoppers and sloped boxes quote the slope from horizontal, so a table entry of 60 degrees means S equal to 30. Same geometry, two conventions, and a 20 degree error if you mix them up. The selector on this page exists for exactly that reason.
Checking it against numbers you already know
If a formula does not reproduce known answers, the formula is wrong. These are the values worth testing against.
| Job | B | S (from vertical) | Miter | Bevel |
|---|---|---|---|---|
| 38 degree crown, square corner | 45 | 38 | 31.62 | 33.86 |
| 45 degree crown, square corner | 45 | 45 | 35.26 | 30.00 |
| Square picture frame, flat | 45 | 90 | 45.00 | 0.00 |
| Hexagon, upright staves | 30 | 0 | 0.00 | 30.00 |
| 4-sided box, sides 15 deg off vertical | 45 | 15 | 14.51 | 43.08 |
| 4-sided hopper, 60 deg slope from horizontal | 45 | 30 | 26.57 | 37.76 |
| 6-sided, 60 deg slope from horizontal | 30 | 30 | 16.10 | 25.66 |
The first two rows are the ones most shops recognise. Thirty-one point six and thirty-three point nine is the pair printed on the side of half the mitre saws sold in the last twenty years, and it is what these formulas give for 38 degree crown on a 90 degree corner. The fourth row is the sanity check at the other end: a hexagonal planter with vertical sides needs no miter at all, just a 30 degree bevel on each edge, which is exactly what stave and barrel work has always done.
Flat frames and upright boxes are the same problem
It is worth seeing that the two familiar non-compound cases are the endpoints of one continuum rather than separate techniques. A picture frame is a set of sides sloped 90 degrees from vertical, which puts all the angle into the miter and none into the bevel: 45 and 0 for a square. A straight-sided box is a set of sides sloped 0 degrees from vertical, which puts all the angle into the bevel and none into the miter: 0 and 45 for a square. Tilt the sides anywhere in between and the angle splits between the two settings, which is what compound means. Watching the two numbers trade off as you drag the slope field is the fastest way to build an intuition for why a sloped planter is harder to cut than either a frame or a box.
Crown moulding, nested versus flat
Crown can be cut two ways and the settings are completely different. Cut it nested, meaning held against the fence at its spring angle upside down as though the fence were the wall and the table were the ceiling, and you need only a miter: half the corner angle, so 45 for a square corner, with the blade upright. That is why nested cutting survives despite being fiddly to hold. Cut it flat on the table, which is the only option once the moulding is too wide to nest on your saw, and you need the compound pair, 31.6 and 33.9 for 38 degree crown at a square corner. This page shows both.
Two things the settings will not tell you. First, inside and outside corners use the same two numbers but opposite hands, and the reliable way to get the hand right is to hold the piece where it will live and mark which end goes long. Second, rooms are not square. A corner that reads 92 degrees is common, and at that angle a square-corner setting leaves a gap you can see from across the room. Measure real corners with an angle finder and use the one-corner mode.
Where the arithmetic stops helping
The formulas assume the joint is a clean plane between two flat faces meeting at a shared edge, which is true for mitred corners in solid stock and is close enough for moulding. They know nothing about how the pieces are held together, and a compound mitre joint in solid wood has end grain against end grain over most of its area, which glues poorly and opens as the wood moves. Splines, keys, dominoes, biscuits or fasteners are what make these joints last; the angles only make them close.
They also assume your saw reads true. Almost none do, out of the box, to the precision these numbers are printed to. Two degrees of error on a square corner is obvious; a tenth of a degree on an eight-sided compound is cumulative across sixteen cuts and will leave you with a gap somewhere. Cut test pieces from the same stock, assemble the corner dry, and trust what you see over what the scale says.
Questions people ask
What are the settings for 38 degree crown moulding on a 90 degree corner?
Cut flat on the table, the miter is 31.62 degrees and the bevel is 33.86 degrees, which saw scales round to 31.6 and 33.9. Cut nested against the fence at the spring angle, it is a plain 45 degree miter with the blade upright. Both are correct and they describe two different ways of holding the same moulding. For 45 degree spring crown the flat settings are 35.26 miter and 30.00 bevel. If your corner is not actually 90 degrees, and most are not, switch to the one-corner mode and enter what you measured.
Is the spring angle measured from the wall or the ceiling?
From the wall, for the purposes of this calculator. Crown described as 52/38 sits at 52 degrees to the ceiling and 38 degrees to the wall, and the number that goes into the slope field with the reference set to vertical is 38. If you only know the ceiling figure, subtract it from 90. If you know neither, hold a scrap of the moulding in place against a framing square and measure the angle its back makes with the wall, or set the reference selector to horizontal and enter the ceiling angle directly.
Why does my saw not go to the bevel angle the calculator gives?
Because compound angles on shallow slopes push the bevel past 45 degrees, and most mitre saws stop at 45 or a little past. A four-sided box with sides only 15 degrees off vertical needs a 43 degree bevel, which just fits; take the slope down to 10 degrees and the bevel is over 44 and you are at the limit. Past that the practical answers are a table saw with a tilted blade and a sled, cutting the complementary angle with the stock reoriented, or building a ramp jig that holds the work at its slope so you can cut it with the settings shown under the in-position section.
How do I find the angle of a corner that is not square?
A digital angle finder laid flat into the corner reads it directly and is the least error-prone method. A sliding bevel gauge set into the corner and then transferred to a protractor works and costs less. The trade method without either is to cut two scraps at a guessed angle, hold them into the corner, and adjust until the joint closes, which is slower but is the only method that is self-correcting. Whatever you use, remember that the two walls may not be plumb either, in which case the corner angle changes with height and you should measure at the height the moulding will sit.
Do the same settings work for inside and outside corners?
The angles are the same; the hand is not. An inside corner and an outside corner at the same wall angle need the same miter and bevel magnitudes, but the pieces are cut on opposite sides of the line, so the long point of the miter is on the back face for an inside corner and the front face for an outside one. Getting this backwards is the most common single mistake in crown work and no calculator can prevent it. Mark each piece in place before it goes to the saw, note which end is long, and cut to the mark rather than to a remembered rule.
What does the included angle between faces tell me?
It is the angle the two finished pieces make with each other where they meet, measured through the material. For a flat square frame it is 180 degrees, because the pieces are coplanar. For an upright square box it is 90. For 38 degree crown at a square corner it is about 112 degrees. It is useful mainly as a check: if you dry-fit the corner and it is visibly a different angle from this figure, something in the setup is wrong before you have cut anything expensive. It is also the angle to set a spline jig or a corner clamp to.