Miter and bevel angle calculator

Two numbers decide whether a frame closes, and people mix them up constantly. The corner angle is what the finished joint measures — 90° on a picture frame, 120° on a hexagon, 135° on an octagon. The miter angle is what you dial into the saw, and it is not that number and not half of it either: saw scales count from square, so an octagon that meets at 135° is cut at 22.5°, while a protractor laid flat on the workpiece across the same cut reads 67.5°. All three describe one cut. Which one is "the miter angle" depends entirely on what is doing the measuring, and a table saw sled, a miter gauge and a miter saw do not agree with each other. The second number appears the moment the sides lean: a planter, a splayed box, a sloping-sided tray. Then one setting is not enough and you need a miter and a bevel at once — what saw manuals call a compound cut. This page gives all of them, plus the thing that actually ruins frames, which is none of the above.

How it is calculated

miter = atan(cos(slope) × tan(180° ÷ sides)) · bevel = asin(sin(slope) × sin(180° ÷ sides))

With the sides upright the slope is zero, the cosine is one and the sine is nothing, so the pair collapses to miter = 180° ÷ sides and no bevel — which is exactly the rule DEWALT prints for its saws: "180º divided by the number of sides equals the miter (if the material is cut vertically) or bevel angle (if the material is cut laying flat)". Lay the same piece flat instead of standing it up and the slope becomes 90°: the miter goes to zero and the whole angle moves onto the bevel, which is the other half of that sentence. Everything between the two is a compound cut. The corner angle is separate geometry: a closed shape turns through 360° in total, so an n-sided frame turns 360°/n at each joint and its corners measure (n − 2) × 180° ÷ n.

Source: DEWALT DWS780 miter saw instruction manual N112900, as published by the Boston University College of Engineering — Cutting Compound Miters — "The chart assumes that all sides are of equal length. For a shape that is not shown in the chart, use the following formula: 180º divided by the number of sides equals the miter (if the material is cut vertically) or bevel angle (if the material is cut laying flat)"

Questions people ask

Is the miter angle half the joint angle or not?

Both, depending on where you put the protractor, and this is the single most common way to cut a frame wrong. Half the joint angle is the angle between the cut face and the long edge of the board: 45° on a square frame, 60° on a hexagon, 67.5° on an octagon. That is what you would mark on the workpiece. A miter saw scale reads zero at a square crosscut, so it wants the complement — 45°, 30°, 22.5° for the same three frames. The square frame is why the confusion survives: 45° is half of 90° and also 90° minus 45°, so the two conventions give the same number and nobody notices until they try a hexagon and cut it at 60° on the saw.

When do I need a bevel as well as a miter?

Whenever the sides do not stand upright. DEWALT defines it plainly: "A compound miter is a cut made using a miter angle and a bevel angle at the same time. This is the type of cut used to make frames or boxes with slanting sides". A flat picture frame is one setting. A planter whose sides lean out, a hopper, a splayed tray or crown moulding cut lying flat on the saw table all need two, and both have to be locked before the blade comes down. The two settings are not independent: change the slope and both move.

What are the settings for crown moulding?

For the North American 52/38 profile, set the side slope on this page to 52° — the top rear angle, the face that lies flat against the ceiling — with four sides for a 90° corner. That gives 31.62° miter and 33.86° bevel, which is exactly what DEWALT stamps into its saws: "special pre-set miter latch points at 31.62º left and right ... and bevel stop pawls at 33.86º left and right". Makita prints the same pair rounded to 31.6° and 33.9° for 52/38° moulding, and 35.3° and 30° for the 45° profile — enter 45° as the slope and you get those too. Two manufacturers, two documents, one geometry.

Why does my frame not close even though every angle is right?

Because a closed frame has two cuts on every piece, so a four-sided frame is eight cuts and an octagon is sixteen, and every one of them is a chance to be a fraction of a degree out. Errors in the same direction do not cancel; they queue up and arrive together at the joint you close last. Half a degree on each of a square frame's eight cuts is four degrees of gap in one corner, which is a visible wedge. That is what the last figure on this page counts. The fix is not more precision on the saw scale but cutting the pairs together, so that whatever the saw is actually set to, both halves of each joint are wrong by the same amount and still meet.

How accurate does the saw setting have to be?

Less accurate than you would think, provided the error is consistent, and far more accurate than the scale allows if it is not. DEWALT is blunt about the crown numbers: "The angles presented for crown moldings are very precise and difficult to set exactly. Since they can easily shift slightly and very few rooms have exactly square corners, all settings should be tested on scrap molding." A hundredth of a degree is not something a detent and a printed scale deliver. Cut two offcuts, hold them together against a known square, and adjust from what you see rather than from what the pointer says.

Does this handle a room corner that is not 90°?

Indirectly. The number of sides here sets the turn at each joint, so four sides means 90° corners, six means 120°, and so on. For a one-off corner that measures, say, 93°, work out how many sides a polygon with that corner would have — 360 ÷ (180 − 93), which is 4.14 — and enter that. The tool rounds the side count to a whole number because a shape cannot have four and a bit corners, so for genuinely out-of-square walls measure the corner, halve it, and take that as the angle between the cut and the board edge instead.

Where do the numbers on this page come from?

The geometry is geometry, but every value checked against it is printed in a manufacturer's manual rather than worked out here. The polygon settings, the compound example (a four-sided box with 26° sides, cut at 42° miter and 18° bevel) and the 52/38 crown presets come from the DEWALT DWS780 instruction manual. The second set of crown figures, including the 35.3° and 30° pair for 45° moulding, comes from the Makita LS1219 manual. Both are linked below, and they agree.

Sources

The documents this page reads its numbers out of, linked so you can check them yourself.

1. DEWALT DWS780 miter saw instruction manual N112900, as published by the Boston University College of Engineering — Cutting Compound Miters — "The chart assumes that all sides are of equal length. For a shape that is not shown in the chart, use the following formula: 180º divided by the number of sides equals the miter (if the material is cut vertically) or bevel angle (if the material is cut laying flat)"
2. Makita LS1219 / LS1219L slide compound miter saw instruction manual — Cutting crown and cove moldings, Table (A), inside corner position (a), 52/38° type — bevel angle Left 33.9°, miter angle Right 31.6°

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