Taper Angle & Sine Bar
Taper angle from two diameters over a length, a ratio 1 : n or an included angle — half angle, taper per unit and per foot — and the sine bar gauge-block stack.
Included and half angle, taper per unit length and per foot, the ratio, and the sine-bar stack for the half and the full angle, with the angle error a rounded gauge-block stack introduces.
Example: Ø30 to Ø20 over 100 is a 1:10 taper, included 5.725°, half 2.862°; a 100 mm sine bar needs a 4.994 stack, and blocks in 0.005 steps put it 0.04 arc-minutes off.
Angle from a taper,
height from an angle.
The taper conventions, and how a sine bar turns an angle into a gauge-block stack.
Taper
A taper is a change of diameter over a length. From two diameters over L, tan(α/2) = (D − d) ÷ (2L) gives the half angle; α is the included angle between the two sides. The same taper expressed as a ratio 1 : n means the diameter changes by 1 per n of length (n = L ÷ (D − d)), and in inches as taper per foot, 12·(D − d)/L. The half angle is what a compound slide or a sine bar is set to.
Sine bar
A sine bar of centre distance C raised on gauge blocks of height h tilts to sin θ = h / C, so the stack for an angle is h = C·sin θ. The page gives the stack for the half angle (turning or grinding the taper) and for the full included angle when it is under 90°. With blocks in fixed steps the stack is rounded, and the page shows the angle the rounded stack actually sets and the error in arc-minutes.
Accuracy
Because dθ = dh ÷ (C·cos θ), a given block error costs more angle as θ grows; above 45° it is usual to set the complement against a square. Standard tapers (Morse, Jacobs, ISO 7:24 and the rest) are not embedded — take D, d and L from your reference. Nothing leaves the browser; the same four anonymous usage counts as the rest of the site apply.
SOURCES
- tan(α/2) = (D − d) ÷ (2L); taper per foot = 12·(D − d)/L; sine bar h = C·sin θ and θ = asin(h / C); standard tapers (Morse, Jacobs, ISO 7:24) are not embedded
Last reviewed 20 September 2026. How results are checked: How we verify.