Woodworking angles and fractions

Inch fractions — type sums like 3' 4-5/16" + 2-7/8" − 1/32", with spaces around operators; × and ÷ (or * and / between spaces) scale a length, and a length ÷ a length gives a ratio. Answers round to the chosen precision with the rounding error shown. Mix metric in freely: 2.5m − 300mm.

Polygon frames — each miter is 180° ÷ sides. With stock width w, the inside edge is outside − 2·w·tan(180°/n). Size by side length, across the corners, or across the flats.

Compound miters — a box of n sides splayed t° from vertical, cut flat on the saw: miter = atan(sin t · tan(180°/n)), bevel = asin(cos t · sin(180°/n)); rip the top and bottom edges at t. A four-sided box at 30° needs 26.6° miter and 37.8° bevel.

Crown moulding — laid flat: bevel = asin(cos s · cos(w/2)), miter = atan(sin s ÷ tan(w/2)) for spring angle s and corner w. 52/38 crown in a square corner: 31.6° miter, 33.9° bevel. Nested upside down, just miter at (180° − w)/2.

Stairs — the fewest risers no taller than the maximum, equal heights, one fewer tread than risers, and the comfort rule 2 risers + 1 tread ≈ 24–26". Planning only — check your building code.

Arcs — radius R = (c²/4 + h²) ÷ 2h from chord c and rise h, the arc angle and length, and heights above the chord to draw the curve with a bent batten.