Taper Calculator — Taper Per Foot and Included Angle

Large Diameter

0.124

Small Diameter

0.0524

Taper Length

0.560

Taper Per Foot

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, 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.

Why Use a Taper Calculator?

Whenever you're machining, verifying, or specifying a tapered part, you need a way to translate raw diameter and length measurements into a standardized taper description that matches your drawing, tooling, or reference standard. Some lathe taper attachments are calibrated in taper per foot; others require a half-angle or included angle setting on the compound slide. Doing this conversion by hand, especially the arctangent step for included angle, is easy to get wrong without a calculator on hand.

This tool computes both values from the same three measurements, so you have whichever figure you need without running two separate calculations.

For related lathe and machining setup work, see the Spindle Speed Calculator to set the correct RPM for turning the taper, or the Thread Calculator if the tapered part also needs a threaded section.

Who Should Use This Calculator?

Machinists turning tapers on a lathe can use the calculator to set up a taper attachment or compound slide angle correctly before starting a cut, and to verify a finished taper against its target specification.

Toolmakers working with Morse or Jarno tapers can use the calculator to check a tool holder or spindle taper against the standard specification for that taper size, by comparing the calculated taper-per-foot value to published reference tables.

Woodworkers and furniture makers shaping tapered legs, spindles, or dowels can use the calculator to plan consistent, repeatable tapers across multiple pieces.

Engineering students and machining apprentices learning taper geometry can use the tool to see how large diameter, small diameter, and length combine into both a rate-based and angle-based description of the same taper.

What Insights Does the Taper Calculator Give You?

Taper Per Foot is the headline result, the diameter change you would see over a standardized one-foot length at the given taper rate, expressed in inches per foot. This is the value used directly on many shop drawings and lathe taper attachment scales. For a taper going from 2 inches to 1.5 inches over a 6-inch length, the calculator returns a taper per foot of 1.0 inch per foot.

Included Angle is the secondary output, the full apex angle of the taper in degrees, useful when your tooling or reference standard is specified by angle rather than by taper-per-foot rate. For the same example, the included angle works out to approximately 4.764 degrees.

Having both values from a single set of measurements means you can move fluidly between taper-per-foot-based and angle-based specifications without needing a separate trigonometric conversion each time.

How to use this Taper calculator

Measure the Large Diameter of the tapered section in inches, using calipers or a micrometer at the wider end.
Measure the Small Diameter in inches at the narrower end of the tapered section.
Measure the Length in inches of the tapered section between the two diameter measurement points.
Read the Taper Per Foot in the highlighted result card, use this value if setting a lathe taper attachment calibrated in inches per foot.
Check the Included Angle shown beneath the headline result, use this value if your tooling or drawing calls for a half-angle or full included angle setting instead.
Compare against a standard taper specification (such as a Morse taper table) if you're verifying an existing tapered part against a known standard.

Show formula & methodology ↓Show less ↑

Formula & Methodology

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

Step 1, Taper Per Foot:

> TPF = ((D − d) ÷ L) × 12

Step 2, Half Angle:

> θ = atan((D − d) ÷ (2 × L))

Step 3, Included Angle:

> A = 2 × θ × (180 ÷ π)

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
- θ = half-angle in radians
- A = included angle in degrees

Worked example, 2 in to 1.5 in over 6 in:

- TPF = ((2 − 1.5) ÷ 6) × 12 = (0.5 ÷ 6) × 12 = 1.0 in/ft
- θ = atan(0.5 ÷ (2 × 6)) = atan(0.04167) ≈ 0.04162 radians
- A = 2 × 0.04162 × (180 ÷ π) ≈ 4.764°

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

What does the Taper Calculator compute?

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.

What is taper per foot and why is it used?

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.

What is included angle and how does it relate to taper per foot?

Included angle is the full angle at the apex of the taper if both tapered sides were extended until they met at a point, 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.

What are Morse tapers and does this calculator apply to them?

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.

Why does the calculator use large and small diameter rather than a taper angle input directly?

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.

How is taper per foot calculated from diameter and length?

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, inches of diameter change per foot of length, that stays consistent regardless of how long the actual tapered section is.

Why does the included angle formula use half the diameter difference?

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.

What's a typical included angle for common tapers?

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.

Can I use this calculator for both internal and external tapers?

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.

How accurate does my diameter measurement need to be for a reliable result?

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.

What industries commonly use taper-per-foot specifications?

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.

How do I set up a lathe taper attachment using taper per foot?

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