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Taper Calculator

Construction

Calculate taper per foot and included angle for a tapered shaft from large diameter, small diameter, and length. Free tool for machinists and toolmakers.

0.124
0.0524
0.560

Taper Per Foot

1
Included Angle
4.772

This calculator computes your Taper Per Foot, Included Angle from the values you enter.

Inputs
Large DiameterSmall DiameterTaper Length
Outputs
Taper Per FootIncluded Angle

What is a Taper?

A Taper Calculator computes two complementary measurements that describe a tapered shaft, bore, or workpiece: taper per foot and included angle. Given the large diameter, small diameter, and length of a tapered section, the calculator returns both figures instantly, giving you the taper described both as a rate of diameter change (taper per foot) and as a geometric angle (included angle).

Tapers appear throughout machining and woodworking โ€” from Morse taper tool holders to tapered furniture legs to tapered pipe threads. Because different trades and reference standards describe taper steepness differently, having both taper per foot and included angle available from the same inputs saves you from doing a second conversion by hand.

How to use this Taper calculator

  1. Measure the Large Diameter of the tapered section in inches, using calipers or a micrometer at the wider end.

  2. Measure the Small Diameter in inches at the narrower end of the tapered section.

  3. Measure the Length in inches of the tapered section between the two diameter measurement points.

  4. Read the Taper Per Foot in the highlighted result card โ€” use this value if setting a lathe taper attachment calibrated in inches per foot.

  5. Check the Included Angle shown beneath the headline result โ€” use this value if your tooling or drawing calls for a half-angle or full included angle setting instead.

  6. Compare against a standard taper specification (such as a Morse taper table) if you're verifying an existing tapered part against a known standard.

Formula & Methodology

The calculator uses standard taper geometry formulas from machinist's reference practice:

Step 1 โ€” Taper Per Foot:

> TPF = ((D โˆ’ d) รท L) ร— 12

Step 2 โ€” Half Angle:

> ฮธ = atan((D โˆ’ d) รท (2 ร— L))

Step 3 โ€” Included Angle:

> A = 2 ร— ฮธ ร— (180 รท ฯ€)

Where:
- TPF = taper per foot in inches per foot
- D = large diameter in inches
- d = small diameter in inches
- L = length of the tapered section in inches
- ฮธ = half-angle in radians
- A = included angle in degrees

Worked example โ€” 2 in to 1.5 in over 6 in:

- TPF = ((2 โˆ’ 1.5) รท 6) ร— 12 = (0.5 รท 6) ร— 12 = 1.0 in/ft
- ฮธ = atan(0.5 รท (2 ร— 6)) = atan(0.04167) โ‰ˆ 0.04162 radians
- A = 2 ร— 0.04162 ร— (180 รท ฯ€) โ‰ˆ 4.764ยฐ

This matches standard machinist's handbook tables relating taper per foot to included angle. For a full picture of the finished part's thread dimensions if it also includes a threaded section, see the Thread Calculator or the narrower Pitch Diameter Calculator.

Frequently Asked Questions

The Taper Calculator computes two related measurements that describe a tapered shaft or workpiece: taper per foot, which expresses how much the diameter changes over a standard one-foot length, and the included angle, which is the total angle formed at the tip of the taper if it were extended to a point. Both are derived from the large diameter, small diameter, and length of the tapered section.
Taper per foot is a standardized way of expressing taper steepness by normalizing the diameter change to a one-foot reference length, regardless of the actual length of the tapered section. It's the traditional unit used on shop drawings and in machinist's handbooks because it lets you compare tapers of different lengths on a common scale, and because many lathe taper attachments are set directly in taper-per-foot units.
Included angle is the full angle at the apex of the taper if both tapered sides were extended until they met at a point โ€” it describes the taper geometrically rather than as a rate of diameter change over length. Included angle and taper per foot describe the same physical taper from two different perspectives; this calculator provides both so you can use whichever your drawing, tooling, or reference standard calls for.
Morse tapers are a standardized family of self-holding tapers used on drill bits, lathe centers, and tool holders, with a taper rate of approximately 5/8 inch per foot (which varies slightly by Morse taper number). You can use this calculator to verify or reverse-engineer a Morse taper's dimensions by entering its large diameter, small diameter, and length, and comparing the resulting taper-per-foot figure against the standard Morse taper specification for that size.
Most real-world measurement scenarios start with diameters and length, since these are what you can directly measure with calipers on an existing part or specify on a new drawing. Working from measured diameters and length to derive taper per foot and included angle mirrors the actual workflow of checking or designing a tapered part in the shop.
Taper per foot equals the difference between the large and small diameters, divided by the length of the tapered section, then multiplied by 12 to normalize the result to a standard one-foot reference length. This produces a rate โ€” inches of diameter change per foot of length โ€” that stays consistent regardless of how long the actual tapered section is.
The half-angle of the taper is measured from the centerline of the shaft to one tapered side, which is why the formula divides the diameter difference by 2 (since diameter spans both sides of the centerline) before taking the arctangent. Doubling that half-angle gives the full included angle, which is the total angle you'd measure across the taper's full width.
A Morse taper has an included angle of roughly 2.9-3.2 degrees depending on the specific taper number, making it a fairly shallow, self-holding taper. A standard 60-degree lathe center has a much steeper included angle by design, since centers are meant to seat and locate rather than grip by friction. Steeper included angles (wider apex angles) generally release more easily, while shallower angles tend to self-lock.
Yes, the geometry formulas are identical whether you're describing an external taper (like a tapered shaft or drill bit shank) or an internal taper (like a tapered socket or bore), as long as you measure the large diameter, small diameter, and length consistently for the taper in question.
Because taper per foot and included angle are both derived from the difference between two diameters, small measurement errors can have a proportionally larger effect on the result than they would on a direct diameter measurement. Use precise measuring tools such as calipers or a micrometer, and measure both diameters at points that accurately represent the true large and small ends of the tapered section.
Taper per foot specifications are common in machine tool work (Morse and Jarno tapers for tool holders), woodworking and furniture making (tapered legs and spindles), plumbing (pipe thread tapers), and general precision machining where a shaft needs to seat into a matching tapered bore with a specific fit and holding force.
Most lathe taper attachments have a scale calibrated directly in inches of taper per foot, so once you know your target taper-per-foot value from this calculator, you set the attachment's guide bar to that value directly. Some lathes instead require setting the compound slide to a half-angle, in which case you'd use this calculator's included angle result divided by two.
Also known as
taper per foot calculatormorse taper angle calculatorshaft taper calculatorincluded angle taper calculatorlathe taper calculator