Convert between large diameter, small diameter and length to get taper per foot, taper per inch, and the half and included angles.
Page last changed 2026-10-19 · based on n/a — standard engineering relationships, no single governing revision
Quick Answer
TPF = (D − d) ÷ L × 12. A taper of 1 in 12 over 12 in means 1 in of diameter change per foot of length. The included angle is 2·atan((D−d) ÷ 2L) — note it needs the half difference, which is the most common error.
Taper and Angle
The Formulas Used
Taper per foot: TPF = (D − d) ÷ L × 12 Taper per inch: TPI = (D − d) ÷ L Half angle: α = atan((D − d) ÷ (2L)) Included angle: 2α
Half Angle vs Included Angle
The single most common taper error is using the full diameter difference where the half difference belongs. The taper angle is formed by the axis and the side of the cone, so the triangle you solve has a base of (D − d)/2 over a length L — not D − d.
The result is that the included angle is twice the half angle, and setting a compound rest to the included angle produces a taper that is visibly wrong. On a lathe the compound rest is set to the half angle, because the tool advances along one side of the cone.
Frequently Asked Questions
How do I calculate taper per foot?
TPF = (D − d)/L × 12, where D is the large diameter, d the small diameter and L the taper length in inches. A taper that changes 0.5 in of diameter over 6 in of length is 1.0 in per foot.
What is the difference between taper angle and included angle?
The taper (or half) angle is between the axis and one side of the cone; the included angle spans both sides and is twice as large. A lathe compound rest is set to the half angle, which is why the two must not be confused.
What is a standard machine taper?
Morse tapers are the commonest, ranging from MT0 to MT6 with roughly 5/8 in per foot of taper. Others include the Brown & Sharpe, Jarno and self-holding tapers. Their angles are all similar — around 1.5° half angle — which is what gives them their self-locking behaviour.
Why does a Morse taper hold without a key?
Because the shallow angle creates enough friction to drive the tool and to resist being pulled out. The half angle is around 1.5°, well below the friction angle for steel, so the taper is self-locking. Steeper tapers need a drawbar or a key.
How do I set a compound rest for a taper?
Set it to the half angle, not the included angle. For a 1 in 12 taper that is atan(1/24) = 2.386°, so the compound is set at 2.386° and the resulting included angle is 4.77°.
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