What makes a cut compound
A cut becomes compound when the joint plane is tilted in two directions at once. Two things put it there: the pieces turn a corner in plan, and they lean out of plumb. Take either one away and you are back to a single-setting cut - a plain miter with a square blade, or a plain bevel with a square fence.
Because both effects act on the same joint plane, they cannot be set independently. Tilting the work changes the miter you need as well as adding a bevel, which is why a splayed four-sided box is not cut at 45 degrees. At a 45 degree splay the miter drops to 35.26 degrees and the blade tilts 30 degrees, and setting the saw to 45 and 45 produces a joint that is open on the inside, open on the outside, or both.
The two inputs are the corner angle C - the interior angle between the two surfaces measured in plan - and the tilt S, how far each piece leans away from vertical. A vertical piece has S = 0. A piece lying flat on the ground has S = 90, which no saw will cut.
If your pieces are plumb and you only need the plan miter, the polygon miter angle calculator is the simpler tool. For crown molding, which is a compound cut with its own conventions, use the crown molding spring angle calculator.
Why the formula uses A and not C/2
The angle that drives both settings is A = 90 − C/2, which is half the change of direction at the corner. Think of walking along one piece and turning onto the other: at a square corner you turn 90 degrees, and half of that is 45. At a hexagon's 120 degree interior corner you turn only 60 degrees, and half of that is 30. A flat hexagonal picture frame is mitred at 30 degrees on the saw, which every frame maker knows, and the formula must reproduce it.
That is the reason to be careful with formulas written as arctan(cos S × tan(C/2)). At C = 90 both expressions give 45, so the error is invisible on square work. At C = 120 one gives 30 and the other gives 60, and only one of them cuts a hexagon.
With A defined, the two settings are:
Miter = arctan(cos S × tan A). Cosine of the tilt is a shrinking factor: at S = 0 it equals one and the miter is simply A; as the work tilts, the miter closes toward zero.
Bevel = arcsin(sin S × cos A). At S = 0 the sine is zero and there is no bevel at all. As the tilt grows, the bevel grows toward S, and the sharper the corner - the larger A - the less of the tilt shows up as blade tilt.
Both formulas come from the same spherical triangle, which is why they share their arguments. You can sanity-check them at the corners of the range: S = 0 gives (A, 0), a plain miter; C = 180 gives A = 0 and the settings collapse to (0, S).
The cut line across the face of your stock is width ÷ cos(miter). That is layout information rather than a saw setting, but it tells you whether your board will fit inside the saw's cutting capacity before you tilt the blade.
Worked example: a four-sided planter with 15 degree sides
You are building a splayed planter box: four sides, each leaning out 15 degrees from vertical, from 5-1/2 in stock.
- Corner angle. Four sides, so C = 180(4−2)/4 = 90 degrees.
- Half the change of direction. A = 90 − 90/2 = 45 degrees.
- Miter. cos 15° = 0.9659258, tan 45° = 1, so the product is 0.9659258 and the miter is arctan(0.9659258) = 44.007 degrees. Set the saw to 44 degrees, a shade under 44-1/8 on a fine scale.
- Bevel. sin 15° = 0.2588190, cos 45° = 0.7071068, so the product is 0.1830127 and the bevel is arcsin(0.1830127) = 10.545 degrees. Tilt the blade to 10.5 degrees.
- Cut line. 5.5 ÷ cos 44.007° = 5.5 ÷ 0.719112 = 7.648 in across the face.
Notice how little the miter moved: from 45.000 to 44.007, barely a degree, for a 15 degree splay. That is the trap in compound work. The miter changes slowly and the bevel changes quickly, so an eyeballed 45 degree miter often looks close enough - right up until the blade tilt is missing and every joint is open on the outside face.
Cut one test joint in scrap at these settings before committing. Two offcuts held together against a square tell you in seconds whether the settings are right, and a compound joint that is a quarter degree out shows clearly at the outside corner.
Reading the settings and cutting them
The miter number is a scale reading measured from a square crosscut, which is how every miter saw is graduated: zero is square, 45 is a picture-frame miter. The bevel number is the blade tilt from vertical, also measured from zero. Neither is a complement, so you set them directly.
Watch the machine's limits. Most compound saws bevel to about 45 or 48 degrees in one direction and miter to about 50 or 60. A sharp corner - a small C - drives A up and can push the miter beyond the scale, and a steeply tilted piece can push the bevel past the head's travel. When that happens, the usual fixes are to cut the piece on its other axis, to build a wedge sled that pre-tilts the work so the saw only has to provide the miter, or to move to a table saw with a tilting arbor and a mitre sled.
Both pieces at a corner get the same two numbers, but mirrored. On a saw that bevels only one way, that means one piece is cut with the face up against the fence and the other with the face down, or the stock is swung to the opposite miter setting. Mark every piece before you cut - which face is out and which end is up - because a compound offcut is very hard to reinterpret afterwards.
Where the corner is measured on site rather than designed, measure the angle properly with a protractor or an angle finder, not by assuming the wall is square. A wall that is 88.5 degrees rather than 90 changes A by three quarters of a degree, which is enough to open a joint in hardwood.
Compound saw settings for common corners
| Tilt S (deg) | 90° miter | 90° bevel | 120° miter | 120° bevel | 135° miter | 135° bevel |
|---|---|---|---|---|---|---|
| 0 | 45.00 | 0.00 | 30.00 | 0.00 | 22.50 | 0.00 |
| 5 | 44.89 | 3.53 | 29.91 | 4.33 | 22.42 | 4.62 |
| 10 | 44.56 | 7.05 | 29.62 | 8.65 | 22.19 | 9.23 |
| 15 | 44.01 | 10.55 | 29.15 | 12.95 | 21.81 | 13.83 |
| 20 | 43.22 | 14.00 | 28.48 | 17.23 | 21.27 | 18.42 |
| 25 | 42.19 | 17.39 | 27.62 | 21.47 | 20.58 | 22.98 |
| 30 | 40.89 | 20.70 | 26.57 | 25.66 | 19.73 | 27.51 |
| 35 | 39.32 | 23.93 | 25.31 | 29.78 | 18.74 | 32.00 |
| 40 | 37.45 | 27.03 | 23.86 | 33.83 | 17.60 | 36.43 |
| 45 | 35.26 | 30.00 | 22.21 | 37.76 | 16.32 | 40.79 |
Every value is arctan(cos S × tan A) and arcsin(sin S × cos A) evaluated with A = 90 − C/2, using the same expressions as the calculator. The 90 degree column at S = 45 gives the well-known 35.26 / 30.00 pair for a fully splayed square box.
Where compound cuts go wrong
- Using half the corner angle instead of half the change of direction. It is right at 90 degrees and wrong everywhere else. A hexagon is mitred at 30, not 60.
- Measuring the tilt from horizontal instead of from plumb. The two are complements, and swapping them turns a 15 degree splay into a 75 degree one.
- Setting the miter and forgetting the bevel. The miter barely moves on gently splayed work, so a joint cut at 45 and 0 looks plausible until it is clamped up.
- Assuming the wall is square. Measure the corner. Two degrees of error opens a visible gap in a mitre.
- Cutting both halves with the same setup. The two pieces at a corner are mirror images; one has to be flipped or swung to the other side of the scale.
- Ignoring the saw's capacity. The cut line across the face is longer than the board is wide, and a tilted blade reduces the depth of cut sharply.
Key terms
- Miter
- A cut across the face of the board at an angle other than square, made by swinging the saw table. On a compound cut it is the setting that turns the corner in plan.
- Bevel
- A cut through the thickness of the board at an angle other than square, made by tilting the blade. On a compound cut it is the setting that accounts for the tilt of the work.
- Splay
- The outward lean of the sides of a box or hopper, measured from plumb. A box with vertical sides has zero splay and needs no bevel.
- Dihedral angle
- The true angle between two planes measured perpendicular to their line of intersection. On a splayed box it is neither the plan corner angle nor the face angle you can measure with a square.
When a compound cut is the wrong answer
Compound miters look precise and are unforgiving. They rely on every piece being exactly the same width, every tilt being identical, and the assembly staying square while the glue sets - and a single tenth of a degree of error is multiplied by the number of corners. For a four-sided box the errors accumulate around a closed loop, so the last joint reveals the sum of the first three.
Where the joint is structural rather than decorative, consider a butt joint with a cleat or a bracket instead. A pergola rafter meeting a beam at an angle can be birdsmouthed or hung rather than compound mitered, and the connection will be stronger and far easier to fit. A sloped fascia at a hip can be scribed rather than calculated, and a scribe automatically absorbs whatever the framing actually did as opposed to what the drawing said.
Where the cut genuinely is compound, cut generously long, test-fit in scrap, and trim. Adjusting a compound cut is a matter of shaving a fraction with the same settings; recutting from scratch at a guessed correction rarely converges.
Related layout tools: the rafter length calculator and the hip and valley rafter calculator handle roof cuts, where the same trigonometry appears under different names, and the board foot calculator prices the stock once the cut list is settled.