Taper Calculator — Angle, Ratio & Tailstock Offset

Every taper question is the same right triangle. The included angle comes from 2 × atan((D − d) ÷ 2L); the ratio form 1:k just says the diameter changes one unit per k units of length, so k = L ÷ (D − d). A 30 to 24 mm taper over 60 mm is 1:10 — a 5.72° included angle. Machinists also quote taper per foot: (D − d) ÷ L × 12, in inches — the same triangle in imperial clothing.

The named tapers are ratios with jobs. 7:24 is the milling spindle family (BT, CAT, SK) — steep at 16.6° included so it releases instantly, because a toolchanger cannot wait for a self-holding taper to let go. 1:20 and shallower are the self-holding family: friction alone drives them, which is precisely why drill-press tooling stays put. The boundary sits near a 7° included angle — steeper releases, shallower grips — and it is worth knowing which side of it your design is on before relying on either behaviour.

The tailstock offset method turns shallow tapers between centres by shifting the tailstock sideways: offset = total length × (D − d) ÷ (2 × taper length). The catch people miss is that the whole-part length matters — a 150 mm shaft with a 60 mm taper section needs 7.5 mm of offset, not 3. It also tilts the centres in their holes slightly, so it is a shallow-taper, non-critical-surface technique; steeper or better-finish work belongs to the compound slide or CNC, where the angle from this page can go straight onto the sine bar for checking.