What is the compound miter formula for a splayed polygon box?

For a box with N sides where the sides lean outward at angle T from vertical: Miter = arctan(cos(T) × tan(180° ÷ N)). Bevel = arcsin(sin(T) × sin(180° ÷ N)). At T = 0° (vertical sides) this reduces to a simple miter of 180°/N with no bevel. The calculator applies this formula live as you type.

What are the crown moulding angles for a standard 90° corner?

For a 90° inside corner using the flat-on-table cutting method. With 38° spring angle: miter 31.6°, bevel 33.9°. With 45° spring angle: miter 35.3°, bevel 30.0°. With 52° spring angle: miter 38.2°, bevel 25.8°. The spring angle is printed on the back of most moulding profiles — check it before calculating.

What miter angle do I need for a 6-sided hexagonal box?

For a 6-sided box with vertical sides: miter 30°, bevel 0°. With sides splayed 10° from vertical: miter 29.6°, bevel 5.0°. With 15° splay: miter 29.2°, bevel 7.4°. Use the polygon box mode in the calculator for any splay angle.

Why is my polygon box assembly rocking or not closing cleanly?

Rocking on a flat surface means the bevel angle is wrong. Gaps on the outside faces of the joints means the miter angle needs adjusting. Adjust one variable at a time, always cutting fresh test pieces from scrap rather than re-adjusting on project material. Also check your saw's bevel scale against a reliable digital angle gauge — the printed scales on most saws are accurate to only about 1°.

Do I need a compound miter saw, or will a standard miter saw work?

You need a compound miter saw (or a table saw with a tilting blade and miter gauge) for any cut that requires both a miter and a bevel angle simultaneously. A standard miter saw whose blade does not tilt cannot make compound cuts. For flat frames where bevel = 0°, any miter saw works fine.

Compound Miter Calculator

Q: How do I find the spring angle of my crown moulding?

The spring angle is usually stamped or printed on the back of the moulding profile, or listed in the product specification. If unmarked, hold the moulding in its installed position against a framing square (acting as wall and ceiling reference) and measure the angle between the moulding's back face and the wall with a digital angle gauge. Common values are 38°, 45°, and 52°.

Get the exact miter angle and bevel angle for your saw — for splayed polygon boxes, flat picture frames, and crown moulding. Plain-English saw instructions included with every result.

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Home › Guides › Math Calculators › Compound Miter Calculator

On this page: Calculator · Which mode to use · Miter vs bevel · The formulas · Reference table · Test cuts · FAQs

Compound miter calculator

For boxes, planters, decorative vessels, and any frame where the sides lean outward. Splay 0° = vertical sides, simple miter only. Splay 10–20° = typical decorative planter.

Number of sides 4 = square · 6 = hex · 8 = octagon Splay angle from vertical (degrees) 0° = vertical · 10–20° = typical planter splay

Quick presets:

Free to use — no sign-up required. Some links on this page are affiliate links — see our disclosure.

Which calculator mode do you need?

The three modes cover every compound miter situation in a woodworking shop. The deciding question is whether your workpiece is tilted in its final position (which requires a bevel) or flat (which does not).

Project type	Mode	Inputs required
Box, planter, tapered vessel with sides leaning outward	Polygon box	Number of sides + splay angle from vertical
Box with vertical sides — square, hex, octagonal	Polygon box, splay = 0°	Number of sides only
Picture frame, mirror, flat panel	Flat frame	Number of sides only — blade stays vertical
Crown, cove, or cornice moulding	Crown moulding	Wall corner angle + moulding spring angle
Hexagonal shelf, clock face, flat polygon	Flat frame, N sides	Number of sides — gives miter only, blade vertical

Miter angle vs bevel angle: what each one controls

These are two separate movements on the saw. Confusing them is the most common reason a compound joint fails at first dry-assembly.

Miter angle — horizontal fence rotation

The miter angle rotates the saw fence (or table) in the horizontal plane. When you set 45°, you are angling the path of the cut across the face of the board. On a miter saw, this is the large primary scale. A standard picture frame uses 45° miter on all four corners with the blade perfectly vertical — that is all that is needed because every piece lies flat.

Viewed from above, the miter angle determines how the end of one piece meets the face of the next around the assembly ring.

Bevel angle — blade tilt from vertical

The bevel angle tilts the saw blade away from 90° to the table. This is the secondary scale, usually on the back of the saw head. Single-bevel saws tilt one direction; double-bevel saws tilt both ways, saving you from flipping pieces end-for-end between cuts on a polygon box.

The bevel controls whether the joint closes cleanly when the assembly is tilted into its final position. A wrong bevel produces a joint that either won't fully close (gap on the outside face) or makes the assembly rock on a flat surface.

Why both are needed for splayed work

A flat frame sits in one plane — only a miter needed. The moment the sides of a box lean outward, the joint face is no longer perpendicular to the plane of the board, so the blade must tilt to match. The miter and bevel angles are linked through the number of sides: you cannot set one independently of the other and expect a tight joint. The calculator outputs both angles from a single formula so you set the saw once and cut.

The compound miter formulas

The calculator uses three distinct formulas — exact trigonometry, not approximations. They match the output of commercial angle calculators and dedicated saw software.

Polygon box with splayed sides

For a box with N sides, sides leaning outward at T degrees from vertical:

Miter = arctan( cos(T) × tan(180° ÷ N) )
Bevel = arcsin( sin(T) × sin(180° ÷ N) )

At T = 0° the bevel drops to zero and miter collapses to exactly 180/N — confirming the formula handles vertical-sided boxes without a special case. At T = 90° (fully flat) the miter drops to zero and the bevel equals 180/N.

Simple flat frame

For a flat N-sided frame where every piece lies in a single plane:

Miter = 180° ÷ N Bevel = 0°

4 sides → 45°. 6 sides → 30°. 8 sides → 22.5°. 12 sides → 15°. Each finished corner of the frame uses two opposing cuts — one on each adjacent piece.

Crown moulding — flat-on-table method

For moulding at a wall corner of C degrees with spring angle S degrees:

Miter = arctan( sin(S) ÷ tan(C ÷ 2) )
Bevel = arcsin( cos(S) × sin(C ÷ 2) )

These angles apply specifically to the flat-on-table method, where the moulding lies flat on the saw table during the cut. This gives more consistent results than the nested (upright against the fence) method because the workpiece is fully supported and cannot shift. Never mix flat-on-table and nested angles — they are different for the same spring angle.

Compound miter quick reference tables

Pre-calculated values for the most common woodworking situations. M = miter angle, B = bevel angle, all in degrees. Use the calculator above for any combination not shown.

Polygon boxes — miter and bevel by sides and splay angle

Sides	Splay 0°	Splay 5°	Splay 10°	Splay 15°	Splay 20°	Splay 30°
4	M 45.0 / B 0.0	M 44.9 / B 3.5	M 44.6 / B 7.1	M 44.0 / B 10.6	M 43.2 / B 14.0	M 40.9 / B 20.7
6	M 30.0 / B 0.0	M 29.9 / B 2.5	M 29.6 / B 5.0	M 29.2 / B 7.4	M 28.5 / B 9.9	M 26.6 / B 14.5
8	M 22.5 / B 0.0	M 22.4 / B 1.9	M 22.2 / B 3.8	M 21.8 / B 5.7	M 21.3 / B 7.5	M 19.7 / B 11.0
12	M 15.0 / B 0.0	M 15.0 / B 1.3	M 14.8 / B 2.6	M 14.5 / B 3.8	M 14.1 / B 5.1	M 13.1 / B 7.4

Flat frames — miter angle only, bevel = 0°

Sides	3	4	5	6	7	8	10	12
Miter	60.0°	45.0°	36.0°	30.0°	25.7°	22.5°	18.0°	15.0°

Crown moulding — 90° inside corner

Spring angle	Miter	Bevel	Notes
38°	31.6°	33.9°	Common on builder-grade and home-centre crown
45°	35.3°	30.0°	Most common profile — easiest to install
52°	38.2°	25.8°	Steeper spring; projects further from ceiling

Crown moulding — 135° bay window corner

Spring angle	Miter	Bevel
38°	14.3°	46.7°
45°	16.3°	40.8°
52°	18.1°	34.7°

For non-standard corner angles (60°, 120°, etc.), outside corners, or any combination not listed, use the crown moulding mode in the calculator.

Test cuts: why they are not optional for compound angles

Compound miter joints are more sensitive to saw calibration error than any other woodworking cut. A saw that is 0.3° off on its bevel positive stop produces acceptable 90° cross-cuts — but that same error on a 22.5° compound cut multiplies across every joint face. An 8-sided box has 16 cut faces; half a degree of bevel error on each accumulates to visible rocking and gaps by the time the ring closes.

The correct process: cut all test pieces from scrap at the same width and length as your final parts, dry-assemble the complete ring with tape or a band clamp, then check for two things. If the assembly rocks on a flat surface, the bevel angle is wrong. If there are gaps on the outside faces of the joints, the miter angle needs adjusting. Adjust one variable at a time — never both together — cutting fresh scrap after each adjustment.

For crown moulding: cut and dry-fit one complete corner before running any full-length pieces. Crown is long and expensive, and a mistake at an outside corner mid-wall cannot be patched. Also confirm the spring angle on your actual moulding before calculating — the stamped number is usually correct, but verify with a digital angle gauge if you have any doubt.

The tools that make this reliable: a digital angle gauge (the printed scales on most saws are accurate to only about 1°), a sliding compound miter saw for crown and multi-sided box work, and a quality band clamp for dry-assembly checks. See the recommended woodworking tools guide for specific suggestions at every budget level.

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Compound miter calculator FAQs

What is the difference between a miter angle and a bevel angle?

The miter angle is the horizontal rotation of the saw fence — it controls the cut angle across the face of the board. The bevel angle is the tilt of the saw blade away from vertical. A flat picture frame needs only a miter angle. Any time the workpiece is also tilted — splayed box sides, crown moulding — both angles are needed simultaneously.

What is the formula for a splayed polygon box?

For N sides and splay angle T from vertical:
Miter = arctan( cos(T) × tan(180° ÷ N) )
Bevel = arcsin( sin(T) × sin(180° ÷ N) )
At T = 0° the bevel disappears and miter = 180/N exactly.

What are the crown moulding angles for a 90° corner?

Flat-on-table method. With 38° spring: miter 31.6°, bevel 33.9°. With 45° spring: miter 35.3°, bevel 30.0°. With 52° spring: miter 38.2°, bevel 25.8°. Check the spring angle printed on the back of your moulding before cutting.

How do I find the spring angle of my crown moulding?

Usually stamped on the back edge of the moulding. If unmarked, hold the moulding in its installed position against a framing square and measure the angle between the back face and the wall with a digital angle gauge. Common values: 38°, 45°, 52°.

My polygon box assembly is rocking. What's wrong?

Rocking = bevel angle error. Outside-face gaps = miter angle error. Adjust one at a time, cut fresh scrap test pieces after each adjustment. Also verify your saw's bevel scale with a reliable digital gauge — most printed scales are accurate to only about 1°.

Do I need a compound miter saw?

You need a compound miter saw (or tilting-blade table saw with miter gauge) for any cut requiring both miter and bevel simultaneously. For flat frames where bevel = 0°, any miter saw works fine.

Can I use these angles on a CNC router?

Yes — the angles are geometrically correct regardless of the tool. For complex compound joinery, tools like Fusion 360 can model and generate toolpaths automatically. See the woodworking design software guide for details.