About

Incorrect angle cuts waste material and produce weak joints. A 45° miter that is off by just 1° creates a visible 0.5mm gap on a 100mm wide board. This calculator computes the precise blade tilt (β) and miter table rotation (α) for both simple and compound angle cuts. It uses standard trigonometric identities applied to the workpiece cross-section. Inputs are material width W, thickness T, the desired joint angle A, and for compound cuts the slope angle S. The tool assumes a flat, rectangular cross-section and does not account for wood grain direction or blade deflection under load.

The calculated cut length L across the face is critical for layout marking. Kerf width k is included to adjust for material removed by the blade. Pro tip: always make a test cut on scrap material. Blade calibration on consumer-grade miter saws can drift by 0.5 - 1.5° from the indicated setting.

Formulas

For a simple miter joint at joint angle A, each piece is cut at the miter angle:

α = A2

The cut length across the face of a board with width W:

L = Wcos(α)

For compound miters where two pieces meet at joint angle A on a slope S, the miter table angle and blade bevel angle are:

α_compound = arctan(sin(S) ⋅ tan(A2))

β_compound = arctan(cos(A2) ⋅ tan(S))

Material removed by the saw kerf along the cut line:

V_kerf = k ⋅ L ⋅ T

Where α = miter table rotation angle, β = blade bevel tilt angle, A = desired joint angle between pieces, S = slope angle from horizontal, W = material width, T = material thickness, k = kerf width of saw blade, L = cut length across the face.

Reference Data

Joint Angle	Miter Setting	Common Use	Cut Length Factor	Notes
30°	15°	Hexagonal frames	1.035	6-sided polygon joint
36°	18°	Pentagonal frames	1.051	5-sided polygon joint
45°	22.5°	Octagonal frames	1.082	8-sided polygon joint
60°	30°	Equilateral triangle frames	1.155	3-sided polygon joint
90°	45°	Picture frames, door casings	1.414	Most common miter joint
120°	60°	Hexagonal planters	2.000	Requires wide blade capacity
135°	67.5°	Octagonal planters	2.613	Near max saw capacity
Regular Polygon Interior Angles
Polygon Sides	Interior Angle	Miter Each Piece	Cut Length Factor	Total Miter Sum
3	60°	30°	1.155	180°
4	90°	45°	1.414	360°
5	108°	36°	1.701	540°
6	120°	30°	2.000	720°
8	135°	22.5°	2.613	1080°
10	144°	18°	3.236	1440°
12	150°	15°	3.864	1800°
Common Kerf Widths by Blade Type
Thin-kerf circular saw		2.0 - 2.4 mm
Standard circular saw		2.8 - 3.2 mm
Table saw (full kerf)		3.0 - 3.8 mm
Hand saw (fine tooth)		0.8 - 1.2 mm
Band saw		0.5 - 1.5 mm
Metal chop saw		2.5 - 3.5 mm

Frequently Asked Questions

Kerf removes material along the cut line. For a 90° miter joint in 100 mm wide stock with a 3.2 mm kerf, the effective shortening per cut is approximately k × tan(α) ≈ 3.2 mm on the long side. If not compensated, two cuts produce a cumulative error of ~6.4 mm in total frame dimension. Always offset your measurement mark by half the kerf width away from the finished side.

Miter angle (α) is the rotation of the saw table in the horizontal plane - it changes the cut direction across the board's face. Bevel angle (β) is the tilt of the blade from vertical - it changes the cut angle through the board's thickness. A simple picture frame uses miter only (α = 45°, β = 0°). Crown molding or sloped joints require both simultaneously (compound miter).

In a compound cut, the geometry is three-dimensional. The miter and bevel interact trigonometrically through sine and cosine projections of the slope. At slope S = 0° (flat), the compound formulas reduce to the simple case: α = A/2, β = 0°. As slope increases, the miter angle decreases while the bevel angle increases. At S = 90° (vertical), the miter goes to 0° and all the angle is in the bevel.

For a regular polygon with n sides, the interior angle is (n − 2) × 180° / n. Each miter cut is half the supplement of that interior angle: α = 180° / n. For example, a hexagon (n = 6) needs α = 30° per cut. This calculator handles this by setting the joint angle A to the interior angle: A = (n − 2) × 180° / n.

Standard miter saws rotate 0° - 50° left and right. Compound miter saws add a bevel tilt of 0° - 48°. This means joint angles above ~100° in simple mode, or certain steep compound combinations, exceed the mechanical range. If your calculated angle exceeds 50° for miter or 48° for bevel, you must flip the workpiece or use a jig. The calculator warns when results exceed common saw limits.

The trigonometry is material-independent. However, grain direction matters for joint strength. End-grain miters in softwoods (pine, spruce) glue poorly - the porous fibers absorb adhesive. Hardwoods (oak, maple) produce tighter joints. Blade deflection in dense hardwoods can shift the effective cut angle by 0.3° - 0.8°. Use a sharp, high-tooth-count blade (80T+) to minimize deflection.