What Is a Taper?
A taper is a gradual reduction in the cross-sectional size of a workpiece along its length, creating a conical profile — similar to a frustum (truncated cone). In machining and manufacturing, tapering means shaping a workpiece so that one end is larger than the other. The larger end is called the major section (major diameter Dl) and the smaller end is the minor section (minor diameter Ds).
Common examples of tapered workpieces include drill chucks, arbors, nails, bolts, screws, and Morse taper tool shanks.
Key Taper Parameters
- Major Diameter (Dl) — The diameter at the larger end of the taper.
- Minor Diameter (Ds) — The diameter at the smaller end of the taper.
- Taper Length (Tl) — The distance between the major and minor sections (not including any straight portions).
- Taper Per Inch (TPI) — The reduction in diameter per inch of taper length.
- Taper Per Foot (TPF) — The reduction in diameter per foot of taper length (TPF = 12 × TPI).
- Taper Angle (θ) — The half-angle between the centreline and the tapered side.
How to Calculate Taper
Taper Per Inch (TPI)
TPI is a dimensionless ratio expressing how much the diameter changes per unit of length:
TPI = (Dl − Ds) / Tl
Taper Per Foot (TPF)
Converting TPI to taper per foot (when diameters and length are in inches):
TPF = 12 × (Dl − Ds) / Tl = 12 × TPI
Taper Angle (θ)
The half-angle of the taper, measured from the centreline to the sloped face:
θ = arctan(TPI / 2)
Conversely, if you know the taper angle, the TPI is:
TPI = 2 × tan(θ)
Metric System
In the metric system, taper is expressed as the ratio of the diameter difference to the length (mm/mm), taper per 100 mm, or taper per meter:
- Taper (mm/mm) = (Dl − Ds) / Tl
- Taper per 100 mm = 100 × Taper
- Taper per meter = 1000 × Taper
Example Calculation
A workpiece has a major diameter of 12 in, a minor diameter of 6 in, and a taper angle of 80.5°. Find the taper length.
- TPI = 2 × tan(80.5°) = 2 × 5.976 = 11.952
- Tl = (Dl − Ds) / TPI = (12 − 6) / 11.952 = 0.502 in
You can verify this directly in the Taper Calculator above by entering the diameters and angle — the calculator fills in the taper length automatically.
Types of Taper
Tapers are classified in several ways:
- By size standard: Jarno, Metric, Brown & Sharpe, Morse, and Jacobs tapers.
- By location: Internal (female) and external (male) tapers.
- By function: Self-holding tapers (steep enough to grip by friction) and quick-release tapers (shallow enough to eject easily).
Ways to Specify a Taper
A taper can be fully defined by any of the following combinations:
- Major diameter + Minor diameter + Taper length
- Major diameter + Minor diameter + Taper angle
- Major diameter + Taper length + TPI or TPF
This calculator supports all of these approaches — simply leave the unknown field blank and the calculator will compute it for you.
Applications of Tapers
- Machine tool spindles — Morse and ISO tapers for drill bits and milling cutters.
- Fasteners — Tapered screws and bolts create self-seating joints.
- Pipes and fittings — Tapered pipe threads (NPT) form leak-proof seals.
- Woodworking — Tapered legs, pins, and joints.
- Automotive — Tapered axle shafts and wheel studs.
Frequently Asked Questions
What is taper per inch (TPI)?
TPI is the change in diameter per inch of taper length. It is a dimensionless quantity because both the numerator (diameter change) and denominator (length) are in inches. A TPI of 0.5 means the diameter decreases by 0.5 in for every inch of length.
What is the difference between taper angle and included angle?
The taper angle (θ) is measured from the centreline to one side. The included angle is the full angle of the cone, equal to 2θ. Our calculator uses the taper angle (half-angle).
How do I convert TPI to TPF?
Multiply TPI by 12: TPF = 12 × TPI. This converts the taper rate from inches per inch to inches per foot.
What is a Morse taper?
A Morse taper is a standardized self-holding taper widely used in machine tool spindles and drill press chucks. There are 8 Morse taper sizes (MT0–MT7) with TPI ranging from approximately 0.050 to 0.063.
Can this calculator work backwards from a taper angle?
Yes. Enter the major diameter, minor diameter, and taper angle — the calculator will compute the taper length for you.