Miter Angle Calculator
Compute the miter angle to cut on each end of a piece in a regular polygon frame so the joints meet flush. Useful for picture frames, hexagonal and octagonal projects, coopered cylinders, and any segmented work where pieces join at equal angles.
Last updated: May 2026
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About this calculator
For a regular polygon (all sides equal length and all interior angles equal), each corner is formed by two pieces meeting at a miter cut, and each piece contributes half of the joint angle. The formula is miterAngle = 90 − (180 / numberOfSides), where numberOfSides is the count of sides in the polygon and the result is the angle to set on your miter saw (measured from a square 90° cut). Variables: numberOfSides must be 3 or more (triangle is the minimum); higher numbers produce angles closer to 90° (a square 90° cut). Geometric derivation: the interior angle of a regular n-gon is (n − 2) × 180 / n degrees; each piece's miter cut is half the supplementary angle = 90 − (180 / n) measured from square. Common values: triangle (3 sides) → 30° miter; square frame (4 sides) → 45°; pentagon (5) → 54°; hexagon (6) → 60°; heptagon (7) → 64.3°; octagon (8) → 67.5°; dodecagon (12) → 75°. Edge cases: most miter saws display the angle from square in one convention; some saws use the complement (the angle from parallel-to-the-fence). For a 67.5° miter, this means a 22.5° saw setting (90 − 67.5 = 22.5). Always confirm your saw's display convention by making a test cut. Cumulative error: small angular errors compound around all n joints — a 0.5° error per joint in an octagon (8 joints, 16 cut faces) produces a total angular error of 4°, which is highly visible as gaps or overlap. Use a digital angle gauge or accurate machinist square to set the saw, and test the setting on scrap before cutting the final pieces. Length accuracy also matters: pieces in the polygon must be exactly equal length, since variation creates angular gaps even when angles are correct.
How to use
Example 1 — picture frame (square, 4 sides). Step 1: miterAngle = 90 − (180 / 4) = 90 − 45 = 45°. Step 2: set miter saw to 45°. Step 3: cut both ends of each of the 4 frame pieces at 45° mirror cuts (one piece has both ends mitered, with the cuts going opposite ways). Verify: a square frame is the standard 45° miter joint — well-known and well-tested. Example 2 — regular hexagon planter or table frame (6 sides). Step 1: miterAngle = 90 − (180 / 6) = 90 − 30 = 60°. Step 2: set miter saw to 60° (or 30° if your saw displays from parallel-to-fence). Step 3: cut both ends of each of the 6 hexagon pieces at 60° mirror cuts. Step 4: dry-fit before gluing; the six pieces should form a perfect hexagon with no visible gaps at any joint. Step 5: use a strap clamp or band clamp around the entire assembly to apply uniform pressure when gluing, since hexagonal joints have no straight reference edge to clamp against like a rectangle. Verify: interior angle of a hexagon = (6 − 2) × 180 / 6 = 120°; each miter contributes half = 60°, matching the cut. Sensitivity check: an octagon (8 sides) gives 67.5°, requiring a more precise saw setting since the angle is less standard.
Frequently asked questions
How do I convert the miter angle from the formula to my miter saw setting?
Most miter saws display the angle measured from 90° (a straight crosscut), so a 45° setting tilts the saw 45° from straight-across-the-board. The formula result is already in this convention: a 67.5° miter for an octagon means set the saw to 67.5°. Some older or specialty saws display the angle from parallel to the fence, where 0° is parallel and 90° is square crosscut; on these saws, set 90 − 67.5 = 22.5°. The simplest check: make a test cut on scrap, measure the cut with a protractor or digital angle gauge, and confirm it matches the formula result. If your saw has detents at common angles (0°, 15°, 22.5°, 30°, 45°), match them to standard polygon values: 22.5° detent = octagon (67.5° from square); 30° detent = hexagon (60° from square); 36° detent = pentagon (54°); 45° detent = square (45°). For non-standard polygons, you'll need to set the saw freely between detents, which requires careful adjustment with a digital angle gauge or fine-tuning fence. Always confirm with a test cut before committing to your final material.
How do I clamp and assemble polygon frames without rectangular clamping references?
Polygon frames have no straight reference edge to clamp against like a rectangular frame, requiring specialized techniques. The most common approach is a strap clamp (also called a band clamp): a long woven strap with a tensioning ratchet that wraps around the entire assembly, applying uniform compressive force inward toward the center. Strap clamps work for any regular or irregular polygon and are essential for octagonal, hexagonal, and circular segmented work. Another approach is dedicated polygon-frame clamping jigs that have triangular or pie-shaped pieces sized to apply pressure perpendicular to each face; these are commercial products designed for specific polygon counts. For very large polygons (8+ sides) or fragile material, build a temporary clamping jig from a piece of scrap material cut to the same polygon shape (slightly smaller) — clamp the workpieces against this inner form. Glue choice: use slow-setting glue (60-minute or longer working time) for complex polygons since assembly takes time. Dry-fit all pieces first to confirm joint quality before applying glue; gluing exposed mistakes is far more costly than re-cutting. For coopered or segmented turning, also use clamping rings made of pipe or wire with multiple tensioning points around the circumference.
Why do polygon frames sometimes have visible gaps even with correct miter angles?
Several factors compound to produce gaps. (1) Saw calibration drift: even quality miter saws can be off by 0.5–1° from the displayed setting; use a digital angle gauge to verify the actual cut angle, not just the saw's scale. (2) Workpiece length variation: if pieces in the polygon vary in length, even by 1/32 inch, the assembly cannot close perfectly; cut all pieces with a stop block to ensure identical length. (3) Cumulative angular error: small per-cut errors multiply across n joints — a 0.25° error per joint in an octagon (8 joints, 16 cuts) produces 4° total error, easily visible. (4) Stock variation: if material isn't perfectly straight (bowing, twisting), the cut angle may be different along the length even with correct saw settings. (5) Cut surface quality: a slightly fuzzy or burned cut from a dull or wrong-type blade produces gaps when pressed against an adjacent piece. (6) Assembly pressure: insufficient clamping force allows joints to spring apart slightly; strap clamps must be tensioned aggressively. (7) Glue squeeze-out preventing close fit: brush off excess glue immediately so it doesn't hold pieces apart. Diagnose by laying pieces face-up on a flat surface with no glue and measuring gaps at each joint — addresses cause specific to that joint.
What are common mistakes when cutting and assembling miter joints?
The most common mistake is using a square 45° angle for any rectangular project assumption — but if you need to build a non-rectangular polygon, you need the formula-derived angle, not 45°. Another error is cutting both ends of a piece at the same angle when they should be mirror cuts (cuts going opposite ways at each end); make sure to flip the workpiece between cuts or use the saw's left/right swing for opposite-handed angles. Confusing the saw's display convention (from square vs. from parallel-to-fence) leads to cutting at the complementary angle and getting pieces that don't fit. Cutting pieces to non-identical lengths (eyeballing or measuring imprecisely) compounds with even correct angles to produce gaps. Failing to test setup on scrap means committing your final material to a flawed setup. Not accounting for the saw kerf when planning piece length: the cut removes ~1/8 inch of material, so the inside-to-inside length of each piece is shorter than the outside-to-outside length by 2× the kerf. Forgetting cumulative-error compounding: a 1° error per joint in a hexagon (6 joints) is 6° total — clearly visible. Using a dull blade produces fuzzy cuts that don't seat properly against mating pieces. Finally, not allowing for wood movement: a tightly clamped rigid frame in dry conditions can spring open in humid conditions; for outdoor or seasonal use, allow some flexibility in glue choice or assembly tolerances.
When should I NOT use this calculator?
Skip this formula for irregular polygons (sides of different lengths, angles different at each corner) — each corner needs to be calculated individually using the specific interior angle at that corner. Do not use it for 3D shapes (pyramidal forms, truncated cones, segmented bowls) where the miter angle in two planes interacts; those need compound miter calculations. Avoid it for joints other than the regular miter — finger joints, dovetails, biscuits, and dowel joints have their own geometry. For decorative segmented turnings on a lathe (segmented bowls), the math is similar but you also need to plan ring stacking and grain orientation, which adds complexity. The formula assumes equal-length sides; if you want unequal sides (a rectangle, parallelogram, or trapezoid), use 45° for the long-and-short rectangle but custom angles for other shapes. For curved-sided polygons (like a frame with curved bow sides), no single miter angle works — that's a coopered or band-built construction. For miter joints in molding profiles where the cross-section shape matters (crown molding, picture frame profiles), the miter is just one cut of a compound miter that also accounts for the molding spring angle; use a crown-molding chart, not just this formula. Finally, for production-scale operations or critical-precision work, invest in a digital angle gauge or angle-verified miter saw rather than relying on the saw's printed scale, which is often imprecise.