Article — Angle Cut & Miter Calculator
Angle cut calculator: miter cuts for any corner
A miter cut joins two pieces of material at a corner. For a regular polygon with n sides, the miter angle is 180° ÷ n on each piece. A square frame (4 sides) needs 45° cuts. A hexagonal table top (6 sides) needs 30° cuts. An octagonal deck (8 sides) needs 22.5° cuts. For rafters and stairs, the angle is arctan(rise/run): a 6:12 roof pitch is 26.57°.
Three situations cover almost every angle cut in woodworking and construction. Cutting a frame, table top, or planter from a regular polygon. Cutting rafters or stair stringers from a rise-to-run ratio. Cutting trim or molding to fit an odd wall corner that isn't quite 90°. Each has its own formula and the same underlying geometry: split the corner angle in half between the two pieces.
What is an angle cut?
An angle cut is any saw cut that is not perpendicular to the length of the workpiece. The angle is measured from the square (90°) position of the saw. A 45° miter cut tilts the saw 45° from square. A 22.5° miter cut tilts it 22.5° from square.
Miter saws have detent stops at the most common angles: 0°, 15°, 22.5°, 30°, 45°. These cover squares, hexagons, octagons, and most picture-frame work. For non-standard angles, you set the saw manually and verify with a digital protractor or test cut.
Miter joints were used in Egyptian carpentry by 1500 BCE. Examples survive in the wooden coffins of pharaoh's burials. The technique has been the same for 3,500 years: split the corner angle in half, cut both pieces, glue and clamp. Power saws made it faster, not different.
The miter cut formula
θ_miter = 180° ÷ n regular polygon, n sidesθ = arctan(rise ÷ run) rafter, stair, rampθ_miter = (180° − α) ÷ 2 odd inside corner αThe polygon formula comes from geometry. A regular polygon with n sides has interior angles of (n − 2) × 180° ÷ n. Each corner is the difference between 180° (a straight line) and the interior angle, which works out to 360° ÷ n. Split that between the two pieces meeting at the corner: 180° ÷ n on each piece.
Miter cuts for regular polygons
The most common shapes and their miter angles:
- Triangle (3 sides) 60° miter (per piece)
- Square frame (4 sides) 45° miter
- Pentagon (5 sides) 36° miter
- Hexagon (6 sides) 30° miter
- Heptagon (7 sides) 25.71° miter
- Octagon (8 sides) 22.5° miter
- Decagon (10 sides) 18° miter
- Dodecagon (12 sides) 15° miter
For a picture frame, all four pieces get cut at 45° miter and the same length on both ends. The cuts must point inward, meeting at the corners. A common beginner mistake is cutting all four pieces with the same orientation; the result is a parallelogram, not a square.
Angle cuts for rafters and stairs
For sloped construction, the cut angle comes from the rise-to-run ratio. Rafters use plumb cuts (vertical) and seat cuts (horizontal) at angles set by roof pitch. A 6:12 pitch means 6 inches of vertical rise per 12 inches of horizontal run. The angle is arctan(6 ÷ 12) = 26.57° from horizontal.
Stair stringers cut at the angle of the staircase, typically 30° to 38° for residential work. Steeper than 38° is too steep to walk down safely; shallower than 28° wastes horizontal space. Building codes specify minimum tread depth and maximum riser height; the angle follows from these.
Angle cuts for odd corners
Real walls are not always 90°. Old houses settle. Plaster wears unevenly. New builds may have intentional non-rectangular shapes. To trim or molding fit a non-standard corner, measure the inside angle with a digital protractor, then apply: θ_miter = (180° − inside angle) ÷ 2.
For an 87° corner (slightly less than square): miter = (180 − 87) ÷ 2 = 46.5°. For a 93° corner: miter = (180 − 93) ÷ 2 = 43.5°. Always cut both pieces at the same miter angle so they meet symmetrically.
Miter vs. bevel vs. compound cuts
Three different angles describe a saw cut. Miter is the horizontal rotation of the saw blade left or right. Bevel is the vertical tilt of the blade left or right. Compound is a cut that uses both miter and bevel simultaneously.
Picture frames use miter only (45° miter, 0° bevel). Crown molding installed at the wall-ceiling junction uses compound cuts. Standard 38° spring crown needs 31.62° miter and 33.86° bevel for inside corners — the geometry is unusual enough that compound miter saws sell specifically for crown molding.
Angle measurements on actual building corners can be off by 0.5° or more. Cut two scrap pieces at your computed miter angle and dry-fit them in the corner. If there is a gap, adjust the saw by half the gap and try again. Two iterations are usually enough to get within 0.1°. Always verify before cutting your final material.
Common angle cut mistakes
The first mistake is calculating the corner angle instead of the miter angle. The corner of a hexagon is 60°, but you set the saw to 30° (half of that). The formula 180° ÷ n gives the miter angle directly; the formula (n − 2) × 180° ÷ n gives the interior angle. Use the right one for what you're cutting.
The second mistake is cutting all pieces with the same orientation. Each piece of a polygon has two cuts that point inward, so each piece flips between cuts. With four pieces in a square frame, you set the saw once, cut all four left ends, then flip the saw to the mirror angle and cut all four right ends.
Practical angle cut tips
Mark the cut line and the keep side. Cut on the waste side of the line, then sand or plane to the line. This protects against blade kerf removing material from the keep piece.
Stack cuts: clamp two or four identical pieces together and cut them at once. You guarantee the angle and length match exactly. Useful for picture frames, hexagonal planters, and any project where multiple identical cuts are needed. Make sure clamps don't interfere with the saw blade path.
For ramps and accessibility, the ADA limits slopes to 1:12 (4.76° from horizontal). UK Approved Document M Volume 1 allows up to 1:12 over a maximum length of 2 m, then a level landing of at least 1.5 m before the next ramp section. The international consensus is that ramps steeper than 5° require handrails on both sides.