Miter angle calculator for polygons and frames

That is 45° for a square, 30° for a hexagon and 22.5° for an octagon. For a single corner at any angle, the miter is (180° − corner) ÷ 2. New to this? Words explained.

The miter formula

A closed polygon turns through 360° in total. Each corner of an n-sided frame turns 360° ÷ n, and each of the two pieces meeting there takes half of that turn.

miter angle = 180° ÷ n        (saw setting from square)
interior angle = 180° − 360° ÷ n
short point = long point − 2 × stock width × tan(180° ÷ n)

Reading your saw’s scale

Most miter saws read 0° for a square cut, so the saw setting is the miter angle itself: 30° for a hexagon. Some table saw miter gauges read 90° at square — on those, set 90° minus the miter angle (60° for a hexagon). A 60° triangle cut is past the stop on many miter saws; cut it with a sled or jig.

Getting tight joints

• Small angle errors multiply. On an octagon there are 16 miter cuts, so a 0.1° error per cut becomes a 1.6° gap at the last joint. Make test cuts on scrap and check a full ring before cutting good stock.
• Cut all pieces with the same stop block so the sides are identical — equal lengths matter as much as the angle.
• For segmented turning rings, the same formula applies: 12 segments per ring means 15° on each end.
• A band clamp pulls every joint closed at once; it is the easiest way to glue up a hexagon or octagon.
• The total stock figure ignores the saw kerf, about 1/8 in per cut. Over a dozen or more cuts that adds up — the same effect as in How many 24 in pieces can you cut from an 8 ft board — so buy a little extra.

Worked examples

The two shapes people build most are on their own pages with cut lengths and stock: What miter angle for a hexagon and What miter angle for an octagon. A four-sided picture frame is the same formula at 45° — How much moulding for an 8×10 picture frame works through one.

Any corner angle

Not every joint is part of a regular frame. Where two pieces meet at a single corner — baseboard or casing around an odd wall corner, a bay window, a sloped trim return — each piece is cut at half of what the corner is short of a straight line. Choose “One corner at any angle” above and enter the corner you measured.

miter angle = (180° − corner angle) ÷ 2
long point − short point = board width × tan(miter)

A square corner gives (180 − 90) ÷ 2 = 45°; a 135° bay corner gives 22.5°; a 100° corner gives 40°. Walls are rarely exact, so measure each corner rather than assuming 90°. Crown molding is different, because it leans off the wall — use the crown molding calculator for that.

If the sides lean outward, a flat miter no longer closes the joint; see Compound miter angles for a 4-sided planter with 15° sides. Crown molding is the same compound problem turned on its side — the crown molding calculator handles it.