How this instrument works
A picture frame, a segmented-turning glue-up ring, or any closed shape built from straight-sided pieces meeting at corners is, mathematically, a regular polygon — a shape with equal sides and equal angles all the way around. Cutting the pieces so they close up into a tight polygon means each corner has to be split evenly between the two boards that meet there, and that split angle is what a miter saw's angle scale actually sets.
The rule is straightforward once you see where it comes from: a full trip around any closed shape is 360°, so a regular polygon with n sides turns through 360°/n at each corner, on the outside. Each of the two boards meeting at that corner contributes half of that turn, so the angle to set the saw to — measured off square, the way miter gauges are marked — is 360° ÷ n ÷ 2, which simplifies to 180° divided by the number of sides. A four-sided frame needs 180/4 = 45° at the saw for each of its four corners; an eight-sided frame needs only 180/8 = 22.5°.
This is distinct from the interior angle of the finished corner itself, which is a larger number — 90° for a square frame, 135° for an octagon — because the interior angle is what the two mitered faces add up to once joined, not the single-board saw setting that produces it. Confusing the two is one of the most common measuring-tape-to-miter-saw mistakes in frame and segmented-turning work, so this instrument reports only the saw-setting figure, the one you actually dial in.
- Enter Number of sides (regular polygon) — 4 for a standard rectangular picture frame, 6 for a hexagon, 8 for an octagon, or however many segments your ring or frame has.
- Read Miter angle per corner (degrees) directly beneath it — the angle to set your miter saw to for every joint in the piece.
- Cut every piece at that same angle; a regular polygon uses one identical angle at every corner, so there is no need to recompute per side.
- For a segmented-turning ring rather than a flat frame, this figure is still the flat miter angle for each segment before any bevel is added for a tapered or bowl-shaped profile.
- The instrument requires at least 3 sides — a triangle is the smallest closed shape a set of straight mitered pieces can form.
Worked example — a standard 4-sided picture frame
Enter 4 into Number of sides (regular polygon) for an ordinary rectangular picture frame. Miter angle per corner reads 45.00 — 180° divided by 4 — the classic 45° setting stamped on the angle scale of almost every miter saw sold.
Cut all four pieces at that same 45° setting, and the two 45° faces at each corner add up to the frame's 90° interior corner once glued together. Swap the input to 8 for an octagon and the same calculation returns 22.5° per corner instead, a much shallower cut for a shape with twice as many sides sharing the same 360° turn.
Questions
Why is the miter angle 45° for a square frame, not 90°?
Because 45° is the angle cut into one board, not the angle of the finished corner. Two boards each cut at 45° meet and add up to the frame's 90° interior corner once joined — the saw setting is always half of what the finished corner angle will be, since two identical pieces share the turn equally.
Does this work for shapes with more than 8 sides?
Yes — the 180°/n formula holds for any regular polygon with 3 or more sides, whether that is a 10-sided decagon (18° per corner) or a 12-sided segmented-turning ring (15° per corner). As the side count grows the miter angle shrinks toward 0°, approaching the flat, gentle cuts used in large circular glue-ups.
Is this the same calculation as a general angle-cut for trim or molding?
The formula is identical — 180° divided by the number of sides — because both come from splitting a regular polygon's corner evenly between two pieces. This instrument is framed around closed shapes like picture frames and segmented-turning rings specifically; a general trim or crown-molding angle-cut instrument applies the same math to running lengths of stock meeting at a wall or ceiling corner.
What angle do I need for an irregular shape, not a regular polygon?
This instrument only covers regular polygons, where every side and every corner is identical. An irregular shape — say, a frame with unequal sides — needs each corner measured and split individually, since there is no single angle that fits all of them the way 180°/n does for a regular shape.
My saw's scale reads from parallel, not from square — do I need to convert?
Most consumer miter saws mark 0° as straight through (a crosscut) and increase the angle from there, which matches how this instrument reports the figure directly. If your saw's scale is marked the opposite way, from a fully swung 90° position, subtract this instrument's reading from 90° to get the setting your particular scale expects.
Does a taller or shorter frame change the miter angle?
No — the miter angle depends only on the number of sides in the polygon, not on the size of the pieces or the frame's overall dimensions. A small 4×6 frame and a large poster frame both use 45° corners as long as both are simple four-sided rectangles.