Taper Calculator

Taper rate: every taper reduces to one number — the diametral taper rate, or how much the diameter changes per unit of axial length: (D − d) / L. Every other value on this page is derived from that single rate: half angle = atan(rate / 2), included angle is twice the half angle, taper ratio is 1 : (1/rate), taper per foot is the rate × 12, and mm per 100 mm is the rate × 100. Because the rate is a ratio of diameter to length in the same units, it converts directly between in/ft, mm/100mm, and 1:N notation without needing the actual size of the part.

Example: a large diameter of 1.500 in, small diameter of 1.250 in, and 2.000 in of taper length gives a rate of 0.125, a half angle of 3.57633° (included 7.15267°, or 3° 34' 34.8"), a ratio of 1 : 8.0000, 1.50000 in/ft, and 12.5000 mm/100 mm. Those are the same values you'll get by entering that geometry above, so you can check the numbers match.

Tailstock offset: to turn a taper between centers by offsetting the tailstock, the offset is half the taper rate times the total distance between centers — offset = rate × centerDistance / 2. This assumes the taper rate is constant over the full distance between centers; if the taper only covers part of a longer workpiece, use the taper's own rate and the full center-to-center distance, not just the tapered section's length. Continuing the example above with 10.000 in between centers gives a 0.62500 in tailstock offset.

Standard tapers: Morse (No. 0–7) and Brown & Sharpe (No. 1–18) each carry their own published, size-specific rate and reference diameters rather than one universal formula — Brown & Sharpe's rate is nominally 0.500 in/ft but published dimensions vary slightly by size, and No. 10 is a well-known exception at 0.51612 in/ft. Jarno tapers are the one family with a true closed-form definition: for Jarno No. N, large diameter = N/8, small diameter = N/10, and length = N/2, which works out to exactly 0.600 in/ft for every size.

Important limitation: these are ideal-geometry relationships for a straight, constant-rate taper. They don't account for tool deflection, setup runout, or an actual part's out-of-round or out-of-straight condition — verify a finished taper by gauge, ring/plug, or blue fit rather than by calculation alone.