Compound Miter Calculator
The bevel formula with cos(180/N) agrees only at four sides and is up to 75 degrees out elsewhere. Verified against 3D geometry.
The formula I had wrong, and why it would have shipped
With the slope measured from horizontal — zero is a flat picture frame, ninety is a vertical box — the two settings are:
tan(mitre) = cos(slope) × tan(180/N)
sin(bevel) = sin(slope) × sin(180/N)
I first wrote the second with cos(180/N), and checked it against a solid built from vectors — two adjacent side planes, the line where they meet, and the plane bisecting them. The mitre matched to fifteen decimal places. The bevel matched only at four sides.
It agrees at four sides because sin 45° and cos 45° are the same number. Everywhere else it is wrong, and across a sweep from three to twenty-four sides the worst error is seventy-five degrees. That is the kind of mistake worth dwelling on: it survives exactly the test everybody runs. Cut a square box, the numbers work, conclude the formula is fine — and then the hexagonal planter falls apart.
Two anchors you can check without any trigonometry
A flat frame is the plain mitre and no bevel at all — 45° for a square, 30° for a hexagon. And a vertical box is the mirror image: no mitre, and a bevel of exactly 180/N.
| Sides | Flat frame | Vertical box |
|---|---|---|
| 3 | mitre 60.0°, bevel 0° | mitre 0°, bevel 60.0° |
| 4 | mitre 45.0°, bevel 0° | mitre 0°, bevel 45.0° |
| 5 | mitre 36.0°, bevel 0° | mitre 0°, bevel 36.0° |
| 6 | mitre 30.0°, bevel 0° | mitre 0°, bevel 30.0° |
| 8 | mitre 22.5°, bevel 0° | mitre 0°, bevel 22.5° |
| 12 | mitre 15.0°, bevel 0° | mitre 0°, bevel 15.0° |
A vertical square box is cut at 45° and a vertical hexagonal one at 30° — anyone who has made either knows that by hand. It is the check that catches the wrong formula by eye, because the wrong version claims 60° for the hexagon.
Crown moulding
Crown sits against the wall at a spring angle, so its slope from horizontal is 90° minus that. The familiar numbers fall straight out:
| Spring angle | Mitre | Bevel |
|---|---|---|
| 38° — the common one | 31.62° | 33.86° |
| 45° | 35.26° | 30.00° |
| 52° | 38.24° | 25.81° |
A 38° spring on a square corner gives 31.62° and 33.86°, which is what is printed on the fence of nearly every mitre saw — and a useful confirmation that the arithmetic is right, since a joiner would spot those being wrong immediately.
Notes
- Measure the corner, do not assume 90°. Rooms are rarely square, and a two-degree error in the corner is a visible gap in the moulding. The corner angle box takes whatever you actually measured.
- Cut a test pair in scrap first. Every saw has some backlash in its detents and every blade some deflection. Two offcuts and a square will tell you more than any calculator.
- Crown can be cut nested instead. Held against the fence at its spring angle, crown is cut with a plain 45° mitre and no bevel — different method, same joint. The compound settings here are for cutting it lying flat.
- Label everything. Compound cuts are handed: the left end of a piece and the right end need mirror settings, and a mislabelled offcut is an expensive way to find that out.
How to use
- Pick an N-sided figure, or switch to crown moulding.
- Set the slope from horizontal — 0 is a flat frame, 90 a vertical box.
- Read the mitre and bevel, and check the two sanity anchors.
- Cut a test pair in scrap before touching the good stock.
Frequently asked questions
What are the compound miter formulas?
With the slope measured from horizontal: tan(mitre) = cos(slope) × tan(180/N), and sin(bevel) = sin(slope) × sin(180/N). The second one is the one that gets written down wrong.
What are the angles for 38 degree crown molding?
A mitre of 31.62 degrees and a bevel of 33.86, on a square corner. Those are the figures printed on the fence of nearly every mitre saw, which is a useful confirmation that the arithmetic is right.
Why do I get the wrong bevel on a hexagonal box?
Almost certainly because the formula you used has cos(180/N) where sin belongs. That version agrees exactly at four sides, because sin 45 and cos 45 are the same number, and is wrong everywhere else — by up to 75 degrees. It survives the one test everybody runs.
How can I sanity check a compound angle?
Two anchors need no trigonometry. A flat frame is the plain mitre 180/N with no bevel at all. A vertical box is the mirror: no mitre, and a bevel of exactly 180/N — 45 degrees for a square box, 30 for a hexagonal one. If a calculator claims 60 for the hexagon, it is using the wrong formula.
What does the slope from horizontal mean?
How far the sides lean from flat. Zero is a picture frame lying in one plane; ninety is a box with vertical sides. Getting this convention backwards swaps the mitre and the bevel, which ruins every cut, so the tool labels both ends of the range.
How do I handle a corner that is not 90 degrees?
Measure it and enter it — the crown mode takes any corner from 30 to 175 degrees. Rooms are rarely square, and a two-degree error in the corner shows up as a visible gap in the moulding.
Should I cut crown moulding flat or nested?
Either works. Held against the fence at its spring angle it is a plain 45 degree mitre with no bevel; laid flat it needs the compound settings here. Nested is quicker for small work and flat is more stable for long lengths.
Why do my compound cuts not match end to end?
Compound cuts are handed — the left end of a piece and the right end need mirror settings. Label every piece before cutting, because a mislabelled offcut is an expensive way to discover it.
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