Miter & Bevel Angle Calculator
Compound miter and bevel angles for frames, bevelled boxes, crown moulding and segmented rings, from one pair of equations rather than four charts.
How to use this calculator
- 1Pick the mode that matches how the work sits, not what it is called - the difference between a bevelled box and a coopered vessel is only the angle the sides lean.
- 2For crown, measure the real corner rather than assuming 90 degrees. Rooms are rarely square and crown shows the error at both edges at once.
- 3Cut a test joint in scrap and dry-assemble the whole ring before cutting good stock.
- 4For segmented work, sand each segment on a jig set once rather than trusting the saw scale for every cut - identical error closes up, random error does not.
How the calculation works
miter = arctan( cos S x tan(E / 2) )
bevel = arcsin( sin S x cos(E / 2) )
E = 360 / n for a regular polygon, or 180 - corner angle for a single corner
S = 0 flat, 90 - spring angle for crown, 90 - lean for a vessel- E
- The angle the work turns through at each joint in plan - the exterior angle, not the corner angle
- S
- How far the face is tilted up from horizontal. Zero for a flat frame, 90 for a vertical-sided box
- miter
- Rotation of the saw from square, with the stock lying flat on the table
- bevel
- Tilt of the blade from vertical, with the stock lying flat on the table
- n
- Number of sides in the polygon
Check the equations against three known cases. A flat square frame has S = 0 and E = 90, giving miter 45 and bevel 0. Standard 38 degree crown in a 90 degree corner has S = 52 and E = 90, giving 31.62 and 33.86 - the published crown figures. A 45/45 crown gives 35.26 and 30.00, also published. If a general formula reproduces all three, it is the right one.
The exterior angle E is what matters, not the interior corner angle. For a twelve-sided ring E is 30 degrees and the flat miter is 15, which is what a segmented turner sets. For a square room corner E is 90 and the flat miter is 45.
Bevel here is measured from a vertical blade with the stock flat. Cutting the same joint with the stock stood on edge exchanges the two angles, which is why staves ripped on edge are set to the complement of the flat bevel.
Worked example
A twelve-segment ring for a turned bowl
- 1.Twelve segments turn through 360 / 12 = 30 degrees at each joint, so the miter is half of that: 15 degrees.
- 2.The work lies flat, so S is zero and the bevel is zero - the blade stays vertical.
- 3.For the ring to contain a 10 in circle after turning, the polygon has to circumscribe it, so each segment’s long side is 10 x tan(15 degrees) = 2.679 in.
- 4.Twenty-four cut faces have to sum to 360 degrees. At a tenth of a degree of error each, the ring closes 2.4 degrees short - which is a visible gap, and why test rings matter.
Result: 15 degree miter, no bevel, 2.679 in segments
Standard crown moulding in a square corner
- 1.52/38 crown sits at 38 degrees to the wall, so its face is 52 degrees up from horizontal.
- 2.A 90 degree corner turns the moulding through 180 - 90 = 90 degrees, so E / 2 is 45.
- 3.Miter is arctan(cos 52 x tan 45) = arctan(0.6157) = 31.62 degrees.
- 4.Bevel is arcsin(sin 52 x cos 45) = arcsin(0.5573) = 33.86 degrees. Both match the figures printed on every mitre saw fence.
Result: 31.62 miter, 33.86 bevel - the standard published crown settings
A flared octagonal planter
- 1.Eight staves leaning 12 degrees out of vertical, so the face is 78 degrees from horizontal and each joint turns 45 degrees.
- 2.Miter is arctan(cos 78 x tan 22.5) = arctan(0.0861) = 4.92 degrees - small, but not zero, and leaving it out is what makes a flared vessel refuse to close.
- 3.Bevel flat on the table would be arcsin(sin 78 x cos 22.5) = 64.65 degrees, which no table saw will do.
- 4.Ripped on edge instead, the setting is the complement: 25.35 degrees, an ordinary cut.
Result: 4.92 degree miter with a 25.35 degree rip bevel, cutting on edge
Four problems that are one problem
Woodworking literature treats picture frames, bevelled boxes, crown moulding and segmented turning as separate topics with separate tables. They are the same geometry seen from different angles, quite literally: the only thing that distinguishes them is how far the face of the workpiece is tilted out of the horizontal plane.
Lay a frame flat and the tilt is zero, so the joint is a plain miter and the blade never leaves vertical. Stand the sides of a box upright and the tilt is ninety degrees, so the miter vanishes and the whole joint becomes a bevel. Crown moulding lives in between, which is precisely why it needs two settings instead of one and why it has a reputation for being difficult.
Seeing it as one problem has a practical payoff beyond elegance. It means a corner that is not ninety degrees, a moulding whose spring angle is not standard, or a vessel with an unusual flare can all be handled directly, rather than by hunting for a chart that happens to cover the case.
Why segmented rings never close
A ring of twelve segments has twenty-four cut faces, and every one of them contributes to whether the ring closes. The angles have to sum to exactly three hundred and sixty degrees, and there is no forgiveness anywhere in the loop - error does not cancel, it accumulates.
The numbers are unkind. A tenth of a degree per cut, which is better than most people can set a mitre gauge and read it, compounds to nearly two and a half degrees of gap across twelve segments. At sixteen segments it is over three. A gap that size is visible from across the room and no amount of clamping pressure closes it without distorting the ring.
What works is making the error identical rather than small. A sanding jig, set once and used for every segment, produces twenty-four faces that are all wrong by the same amount, and a ring of identically wrong segments closes perfectly - the errors cancel because the shape is still regular. This is why serious segmented turners sand rather than saw the final surface, and why they cut a test ring from scrap before touching the good stock.
- Cut a test ring — Dry-assemble it with a band clamp and measure the gap before cutting anything you care about.
- Sand to a jig — Identical error closes; random error does not. A disc sander with a fixed fence beats a saw scale.
- Half the segments, twice — Glue up two half-rings, then true the two mating faces flat. Two joints to fix instead of twelve.
- Watch the blade, not the scale — Fence slop and blade drift are usually larger than the graduation you are trying to read.
Crown moulding, and the method that avoids all of this
Crown is the case that drives people to compound mitre saws, and it is worth knowing that the compound setting is optional. If the moulding is held against the saw fence at exactly the angle it will sit on the wall - upside down, with the two flats bearing on the fence and the table - then the saw only needs an ordinary miter, at half the corner turn. No bevel at all.
That method predates compound saws and remains the fastest way to cut crown accurately, provided the moulding is genuinely held at its installed angle every time. Purpose-made crown stops clamp to the saw and guarantee it; a strip of wood screwed to the fence does the same job for nothing.
The compound settings exist for wide crown that will not fit nested, and for anyone cutting it flat on a large saw. They are also unavoidable on non-standard corners, where the published tables run out. The main practical warning is that rooms are not square: a corner that reads ninety on the plan is routinely a degree or two out, and crown displays that error at both edges of the joint simultaneously. Measure the corner, do not assume it.
What this assumes, and where it stops
Assumptions
- The polygon is regular - all sides equal and all angles equal. Irregular shapes need each corner calculated separately using the corner mode.
- Angles are for stock lying flat on the saw table unless the result explicitly gives a rip bevel for stock on edge.
- Segment sizing assumes the polygon circumscribes the finished circle, so there is material to turn away at the flats.
- Saw scales are assumed to be accurate. In practice they are the largest source of error in the whole process.
Limitations
- It gives the geometry, not the machine. Many compound settings - particularly for steeply tilted faces - exceed what a saw can physically reach, which is why the rip-on-edge alternative is reported.
- Blade kerf is not accounted for in the segment lengths. Cut long and trim, or add a kerf per cut when laying out stock.
- Wood movement will open and close segmented joints seasonally, and a ring glued up from long-grain and end-grain segments moves unevenly around its circumference.
- Irregular polygons, compound curves and mouldings that are not straight in plan are outside what these equations describe.
Common questions
What angle do I set for a hexagonal frame?
Thirty degrees. A six-sided frame turns through 360 / 6 = 60 degrees at each joint, and lying flat the miter is half of that. The general rule for any flat regular polygon is 180 divided by the number of sides: 45 for a square, 30 for a hexagon, 22.5 for an octagon, 15 for a twelve-sided ring.
Why do my crown moulding corners have a gap at the top but not the bottom?
The bevel is wrong rather than the miter. Those two settings control different things: the miter governs whether the joint closes in plan, and the bevel governs whether it closes through the thickness. A gap that varies from the top of the profile to the bottom points at the bevel, and usually at the spring angle having been entered from the wrong reference - 38 rather than 52 or the reverse.
Can I cut crown without a compound mitre saw?
Yes, and it is often better. Hold the moulding against the fence at exactly the angle it sits on the wall - upside down, with both back flats bearing - and the cut becomes an ordinary miter at half the corner turn, with the blade vertical. A crown stop or a strip of wood on the fence keeps the angle consistent, which is the only hard part.
My segmented ring will not close. What did I do wrong?
Probably nothing, in any single cut. Twenty-four faces at a tenth of a degree each is 2.4 degrees of accumulated gap, and that is a good tenth of a degree. Fix it by making the error consistent instead of small: sand every segment on the same jig with the same stop, so all twenty-four faces are identical. A regular polygon of slightly-wrong segments still closes.
What does the "rip bevel" figure mean?
It is the same joint cut with the stock standing on edge against the fence rather than lying flat on the table. The two orientations exchange the roles of miter and bevel, so a joint that needs an impossible 67.5 degree blade tilt when cut flat becomes an ordinary 22.5 degree rip when cut on edge. Staved and coopered work is nearly always cut this way for exactly that reason.
Sources
- Compound miter geometry and crown moulding angles — Fine Woodworking
Formula and content last reviewed on .
Results are estimates for information only, not professional advice.
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