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°.

Taper Calculator

Convert between large diameter, small diameter and length to get taper per foot, taper per inch, and the half and included angles.

Data verified 2026-09-29 · 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

Large diameter (in)

Small diameter (in)

Length of taper (in)

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°.

Value Sources

Each data column on this page is tied to the source it came from. The numbers in square brackets correspond to the table headers above.

#	Source	Type	Revision / method
[1]	ASME B1.1 — Unified Inch Screw Threads	standard	ASME B1.1-2019 — source
[2]	ASTM A615 — Deformed steel bars for concrete reinforcement	standard	ASTM A615/A615M-20 — source
[3]	ASTM E140 — Hardness Conversion Tables	standard	ASTM E140-12b — source
[4]	Values computed in your browser	derived	Evaluated locally from the formulas shown on the page. No data leaves the device.
[5]	ISO 4287 — Surface texture: Profile method	standard	ISO 4287:1997 — source
[6]	ISO 68-1 — Basic profile	standard	ISO 68-1:2023 — source
[7]	NFPA 70 NEC Table 310.16	standard	NEC 2023 (NFPA 70-2023) — source

Data Sources

Standard	Revision	What it covers on this page
Formulas as shown on this page	n/a — standard engineering relationships, no single governing revision	every value this calculator produces

Cross-checked against:

Formulas verified against the corresponding reference chart on this site, so the calculator and the published tables agree

Derived values — the following values on this page are calculated, not taken directly from the standard:

Value	How it is derived
All outputs	Computed in the browser from the formulas above. No data leaves the device.

Outputs are computed from the formulas shown. Verify against the governing standard for design or acceptance work.

Accuracy and use. The values on this page are compiled from the published standards and cross-checked sources listed above. Where values are derived, the derivation is stated. No warranty, express or implied, is made as to the accuracy or completeness of this information, and no liability is accepted for any loss or damage arising from its use. Engineering reference data is provided for guidance in preliminary work — before a value is used for design, fabrication or acceptance testing, verify it against the current revision of the governing standard and against your own inspection. The user assumes all risk and responsibility in connection with the use of this information.

Cite This Page

Every value on this page is traceable to the sources listed above. If you use the data in a document, paper or report, cite it as:

Plain text

Engineering Reference. "Taper Calculator." https://refmanual.com/calculators/taper-calculator/ (accessed 2026-09-29). Data from Formulas as shown on this page, n/a — standard engineering relationships, no single governing revision.

BibTeX

@misc{taper-calculator,
  title        = {Taper Calculator},
  author       = {{Engineering Reference}},
  howpublished = {\url{https://refmanual.com/calculators/taper-calculator/}},
  note         = {Accessed: 2026-09-29},
  source       = {Formulas as shown on this page, n/a — standard engineering relationships, no single governing revision}
}

RIS (EndNote, Zotero, Mendeley)

TY  - ELEC
TI  - Taper Calculator
AU  - Engineering Reference
PB  - Engineering Reference
UR  - https://refmanual.com/calculators/taper-calculator/
PY  - 2026
DA  - 2026-09-29
N1  - Source: Formulas as shown on this page, n/a — standard engineering relationships, no single governing revision
ER  -

Each row in the tables above also has a permanent link — hover a row and use the # link to cite a single value rather than the whole page.