Miter Joint Angles for Equal or Unequal Board Widths

Formula and method

Equal-width boards meeting at90 degrees have45-degree geometric seam angles and45-degree saw-setting magnitudes.

For wA≤wB, α=atan2(wA sinθ,wB+wA cosθ), β=θ−α. Swap board roles for wA>wB; saw magnitude=|90°−geometric angle|.

Worked example

Enter these known values and leave the other values blank.

Assumptions and limitations

• Widths are positive perpendicular board widths measured in the same plane. The included angle is strictly between0 and180 degrees. Zero-width and parallel/fully folded degeneracies are excluded.
• The seam joins the common inner corner to the intersection of the two offset outer board edges. The two geometric angles sum to the included angle. Unequal seam angles use atan2 and stable half-angle forms near a right or folded joint; the wider angle is the remaining part. Equal-width seam and saw angles are evaluated directly as half the included angle and half its supplement.
• Tiny finite widths and included angles remain in the same strict positive domain and use scientific display when ordinary relative formatting cannot represent them. Literal width differences retain their decimal digits through exact international length conversion; diagram and note branches compare exact signs. Angle conversion retains extra decimal digits against the existing64-digit pi constant. Other accepted math expressions use the standard calculator parser and precision. No physical lower bound is added.
• Saw settings are nonnegative magnitudes measured from a square crosscut. A left/right direction, stock orientation, bevel, blade kerf, compound cut or machine adjustment is not determined. Test the geometry and actual equipment setup separately.
• All main width controls and helper unit alternatives remain. Converted-width helpers now represent the same physical length, defaulting to exact international inches. Notes print the actual selected units.
• Clearing a width leaves that dimension unknown; the captured linear converted-width inverse remains available. One-way trigonometric seam outputs retain their forced-output behavior when all physical inputs are known and valid; clearing a required input withholds those results. The nine conditional static diagrams show the relationship and angle branch without scaling to input.

Common questions

References

• OpenStax: planar trigonometric definitions
• NIST: exact inch and international lengths
• Calculation definition and unit reference

Preserve retained decimal input and exact finite length factors before width comparisons; retain extra angle-conversion coefficient digits without changing global precision. Represent converted board-width helpers as the same physical length as the board inputs. Use stable right/obtuse half-angle identities and atan2 for unequal seams and signed complements; evaluate equal-width seam and saw angles directly as theta/2 and (pi-theta)/2, including every print variant. Preserve captured output precision and withhold trigonometric results when physical inputs are absent or outside the captured open domain. Select inches initially for the hidden converted-width helpers in both preferences, preserving every unit alternative and using dynamic unit labels in the authored notes. Enforce the captured strictly positive board widths and included angle below pi with exact signs, retaining the original open domain without a physical lower bound. Reuse the established full-schema source-pinned scientific display fallback for finite tiny inputs.

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