How this instrument works
The cut angle for a regular n-sided shape is 180 divided by n. Every straight piece framing a polygon — a hexagonal planter box, an octagonal window casing, a five-sided garden bed — meets its neighbor at the polygon's interior angle, and a symmetric miter splits that meeting point into two equal cuts. Set a miter saw to 180/n degrees off square and cut every piece the same way; the corners close cleanly all the way around.
This is the same setting carpenters mean when they say 'cut an octagon at 22.5 degrees' or 'a hexagon closes at 30 degrees' — 180/8 = 22.5 and 180/6 = 30 exactly. The formula generalizes both familiar numbers and every less-common shape in between: a pentagon closes at 36 degrees, a decagon at 18, and so on, all from the same single division.
The relationship comes straight from polygon geometry: a regular n-sided polygon's exterior angle is 360/n, and a symmetric miter cuts each piece at half of that exterior angle — 180/n. Because every regular polygon's exterior angles sum to exactly 360 degrees no matter how many sides it has, this one formula covers triangles, hexagons, decagons and anything else with equal sides and equal angles.
This instrument covers the general case — any regular-sided frame, box, planter or trim run. A picture frame or a segmented-turning lathe blank uses the identical 180/n arithmetic, and this site's dedicated miter-angle instrument frames that same math around those two specific crafts if that's the project you're working from.
- Enter the shape's side count into Number of sides (regular polygon) — 6 for a hexagon, 8 for an octagon, and so on.
- Read Cut angle per corner (degrees) beneath it — that is the angle to set on a miter saw or protractor, measured off a square cut.
- Cut every piece framing the shape at the same angle; a symmetric miter closes each corner using two matching cuts.
- Enter at least 3 sides — a shape needs at least a triangle's worth of sides before a cut angle means anything.
Worked example — a 6-sided hexagon frame
Enter 6 into Number of sides (regular polygon) for a hexagonal planter box or trim frame. Cut angle per corner (degrees) reads 180 / 6 = 30.00 degrees — set the miter saw to 30 degrees off square and cut all six sides identically.
Each hexagon corner is formed by two boards meeting; because each board is mitered 30 degrees off square, the two cut faces together account for the difference between a straight 180-degree line and the hexagon's 120-degree interior angle, closing the joint with no gap.
Questions
Why does the cut angle equal 180 divided by the number of sides?
Because a regular polygon's exterior angles always sum to exactly 360 degrees, so each one measures 360/n. A symmetric miter joint splits that exterior angle in half between the two meeting pieces, giving 180/n per piece — the same arithmetic behind the familiar '22.5 degrees for an octagon' and '30 degrees for a hexagon' rules carpenters already use.
What angle do I cut for an octagon?
22.5 degrees off square. Enter 8 into Number of sides (regular polygon) and Cut angle per corner (degrees) reads 180/8 = 22.5 exactly — the standard setting for an eight-sided picture frame, gazebo plate, or window surround.
Does this work for irregular shapes too?
No — the formula assumes a regular polygon, where every side is the same length and every interior angle is identical. An irregular shape has a different angle at each corner, so each joint would need to be measured and cut individually rather than read off one shared number.
Is this the same calculation as the miter-angle instrument?
It uses the identical 180/n arithmetic, yes — but this instrument is framed for general trim, molding, boxes and planters built around any regular polygon, while the dedicated miter-angle instrument frames that same math specifically around picture frames and segmented-turning lathe blanks. Either one returns the correct angle; pick whichever framing matches the project in front of you.
What's the minimum number of sides this handles?
Three. A three-sided shape is the smallest closed regular polygon possible, and the instrument requires at least that many sides before it will compute a cut angle — anything smaller isn't a valid polygon to miter around.
How do I actually set this angle on a miter saw?
Most miter saws are calibrated so 0 degrees is a square (90-degree) crosscut; rotate the saw's angle gauge to the computed cut angle off that zero point, in whichever direction the piece needs to lean. Always cut a test piece from scrap first and dry-fit two corners before committing full-length stock, since saw calibration drifts slightly over time.