Taper Calculator

Q: What angle do you set the compound rest for a taper?

Set the compound rest to the half angle , not the full included angle. The compound cuts one side of the part, and the lathe spins the other side into the same cut. If a print says 10° included, you swing the compound to 5°. Using 10° would make the taper twice as steep as you want.

Q: What is the difference between a self-holding and a self-releasing taper?

A self-holding taper is shallow, around 5/8 inch per foot or less. Friction alone locks the tool in place. Morse and Brown & Sharpe tapers work this way. A self-releasing taper is steep, like the 7:24 spindle taper (about 3.5 inch per foot). It will not stick, so it needs a drawbar or retention knob to hold it. It also drops out fast for quick tool changes.

Taper Calculator

Introduction

A taper is a part that gets wider at one end and thinner at the other. Machinists make tapers on lathes, grinders, and mills every day. This calculator works out taper geometry: diameters, length, taper rate, and angle.

Type in any three of these four values: the large end diameter (D), the small end diameter (d), the length (L), and the taper rate (T). The tool finds the missing one. You can also enter the included angle in degrees, minutes, and seconds instead.

Once you hit Calculate, the taper is shown in every form you might need: taper per foot, taper per metre, taper per inch, percent, ratio, included angle, and half angle. Step-by-step math is shown so you can check the work or learn the formula. A scaled drawing of your taper is plotted too.

There is also a sine bar calculator for surface grinder setups. It tells you the tilt angle and the exact shim or gauge block stack you need, either along the length of your part or across its width.

Built-in charts list standard tapers, Morse tapers (MT0 to MT6), Brown & Sharpe tapers, and fraction-to-decimal values. The row that matches your taper closest is marked for you. Works in both inch and metric units.

How to use our Taper Calculator

Enter any three of these four values (large end diameter, small end diameter, length, and taper rate) and the calculator solves the missing one. It then shows your taper per foot, per metre, per inch, as a percent, as a ratio, and as an included angle and half angle, plus a step-by-step solution, a scale drawing, and sine bar shim sizes for grinding setups.

Unit System: Pick Imperial (inches) or Metric (millimetres). Any numbers you already typed are converted for you.

Large End Diameter (D): Type the width across the wide end of the taper, then pick its unit. Leave it blank if you want the calculator to find it.

Small End Diameter (d): Type the width across the narrow end of the taper, then pick its unit. Leave it blank to solve for it.

Length (L): Type the distance along the part between the two diameters you measured. Leave it blank to solve for it.

Taper Rate (T): Type how much the diameter changes per foot, per metre, or per inch. Leave it blank to solve for it.

Degrees (°): If you know the included angle instead, type the whole degrees here and the taper rate fills in for you.

Minutes (′): Add the minutes part of your angle, from 0 to 59.

Seconds (″): Add the seconds part of your angle, from 0 to 59.

Plot units: Choose the unit used on the taper profile chart, so the drawing matches how you measure.

Small End (thinnest): In the sine bar section, type the thinnest finished thickness of your part.

Large End (thickest): Type the thickest finished thickness at the other end or edge.

Length of Taper / Width of Taper: Type the distance between those two thickness points. Use the Length box for a taper along the part, or the Width box for a taper across the part.

Sine Bar Length: Type the centre-to-centre roll spacing of your sine bar, such as 5 inches. The shim stack size is worked out from this.

Shim Thickness unit: Pick the unit you want the gauge block stack shown in, then press Calculate.

What Is a Taper?

A taper is a part that gets wider at one end and thinner at the other. Think of a cone, a drill shank, or a wedge. Machinists use tapers to lock tools into machines, to make parts self-centering, and to help parts slide together and grip tight.

The Four Key Numbers

D is the large end diameter (the thick end).
d is the small end diameter (the thin end).
L is the length between those two diameters.
T is the taper rate, or how fast the size changes.

If you know any three, you can find the fourth.

Taper Formulas

Taper per inch = (D − d) ÷ L
Taper per foot = (D − d) ÷ L × 12
Taper per metre = (D − d) ÷ L × 1000
Half angle = arctan[(D − d) ÷ (2 × L)]
Included angle = 2 × half angle

Half Angle vs. Included Angle

The included angle is the full angle between both sloped sides. The half angle is just one side, measured from the center line. This matters a lot on a lathe: you swing the compound rest to the half angle, not the full angle. Using the wrong one doubles your error.

Ways to Write a Taper

The same taper can be written many ways, and prints use all of them:

Inches per foot, common in U.S. shops (example: 0.600 in./ft.).
Millimetres per metre, common on metric prints.
Ratio, like 1:20, meaning the size grows 1 unit for every 20 units of length.
Percent. A 5% taper grows 0.05 per 1 of length.
Degrees, written as degrees, minutes, and seconds (like 2° 51′ 51″).

Standard Tapers

Many tapers are set by industry standards so tools from different makers fit the same machines.

Morse taper (MT0 to MT6), used on drill shanks, lathe tailstocks, and spindles. Near 5/8 inch per foot, so it holds by friction alone.
Brown & Sharpe (#1 to #18), about 1/2 inch per foot, common on older milling machines and collets.
Common ratios of 1:20, 1:10, 1:5, and others show up on pins, valves, and machine spindles.

Matching your taper to the closest standard helps you pick off-the-shelf tooling instead of making a one-off part.

Sine Bars and Shims

On a surface grinder, you can't just spin a taper. Instead you tilt the part with a sine bar. A sine bar is a hardened bar with two rolls set an exact distance apart, usually 2, 3, 5, or 10 inches. You stack gauge blocks under one roll to tip it to the angle you want.

The math is simple:

Angle = arcsin[(thick end − thin end) ÷ span]
Shim stack = sine bar length × sin(angle)

A length taper runs along the long side of the part. An edge or width taper runs across the short side. Same math, different direction. Just be sure you use the right span, or the part comes out wrong.

Shop Tips

Always check whether the print gives the included angle or the half angle.
Measure D and d at the exact points that L spans, or your taper rate will be off.
Keep gauge blocks and sine bar rolls clean. A speck of dirt can throw off your angle.
Small angle errors grow over length. On long tapers, tight tolerance on the angle matters more than on the diameters.
Self-holding tapers (like Morse) grip by friction. Self-releasing tapers (like steep 7:24 spindle tapers) need a drawbar.

Formulas used

Taper rate from diameters and length

T = \frac{D - d}{L}

Large end diameter from taper

D = d + T \times L

Length of taper

L = \frac{D - d}{T}

Taper expressed per foot, per metre and as a percentage

T_{ft} = \frac{D-d}{L} \times 12, \quad T_{m} = \frac{D-d}{L} \times 1000, \quad T_{\%} = \frac{D-d}{L} \times 100

Taper ratio

1 : \frac{L}{D - d}

Half angle and full included angle of taper

\frac{\alpha}{2} = \arctan\!\left(\frac{D-d}{2L}\right), \quad \alpha = 2\arctan\!\left(\frac{D-d}{2L}\right)

Taper rate from included angle

T = 2\tan\!\left(\frac{\alpha}{2}\right)

Sine bar angle and shim (gauge block) stack height

\theta = \arcsin\!\left(\frac{t_{large} - t_{small}}{S}\right), \quad \text{Shim} = L_{bar} \times \sin\theta

Frequently asked questions

What is a Morse taper used for?

A Morse taper holds tools in machines by friction. You find it on drill bits, reamers, lathe centers, and tailstocks. The tool slides in and the taper grips it tight with no bolts needed.

Morse sizes run from MT0 (smallest) to MT6 (largest). Each size has its own diameter and length, so a MT2 drill only fits a MT2 socket.

How do you find the taper angle of a shaft?

Measure the big diameter (D), the small diameter (d), and the length (L) between those two spots. Then use:

Half angle = arctan[(D − d) ÷ (2 × L)]
Included angle = 2 × half angle

Example: D = 1.5 in., d = 1.0 in., L = 6 in. Half angle = arctan(0.5 ÷ 12) = 2.386°. Included angle = 4.77°.

What angle do you set the compound rest for a taper?

Set the compound rest to the half angle, not the full included angle. The compound cuts one side of the part, and the lathe spins the other side into the same cut.

If a print says 10° included, you swing the compound to 5°. Using 10° would make the taper twice as steep as you want.

What does a 1:20 taper mean?

It means the diameter grows 1 unit for every 20 units of length. So over 20 inches of length, the diameter grows 1 inch.

That works out to 0.05 inch per inch, or 0.6 inch per foot, or a 5% taper. The included angle is about 2° 52′.

How do you convert taper per foot to degrees?

Divide the taper per foot by 12 to get taper per inch. Then:

Half angle = arctan(taper per inch ÷ 2)
Included angle = 2 × half angle

Example: 0.600 in./ft. ÷ 12 = 0.05 in./in. Half angle = arctan(0.025) = 1.432°. Included angle = 2.86°, or about 2° 52′.

How do you calculate gauge blocks for a sine bar?

Multiply the sine bar length by the sine of the angle you want.

Shim stack = sine bar length × sin(angle)

Example: a 5 inch sine bar set to 3° needs 5 × sin(3°) = 0.2617 inch of gauge blocks under one roll.

What size sine bars are standard?

The most common sine bars are 5 inch and 10 inch center-to-center. Small ones come in 2.5 inch and 3 inch. Metric bars are usually 100 mm, 200 mm, or 250 mm.

Longer bars give finer control at small angles. Shorter bars fit small parts and tight spaces.

What is the difference between a self-holding and a self-releasing taper?

A self-holding taper is shallow, around 5/8 inch per foot or less. Friction alone locks the tool in place. Morse and Brown & Sharpe tapers work this way.

A self-releasing taper is steep, like the 7:24 spindle taper (about 3.5 inch per foot). It will not stick, so it needs a drawbar or retention knob to hold it. It also drops out fast for quick tool changes.

Why is a Morse taper not exactly 5/8 inch per foot?

The Morse standard was set in the 1860s and each size was fitted by hand and by trial. The sizes were never made to one exact number.

Real Morse tapers run from about 0.5986 in./ft. (MT1) to 0.6315 in./ft. (MT5). All are close to 5/8 inch per foot but none match it exactly. That is why you must use the right numbers for each size, not a rounded value.

How do you measure a taper without a gauge?

Use two matched precision balls or pins of known size. Set one pair in the taper, measure the height with a mic or height gauge, then repeat with a spacer under them at a known distance.

The change in diameter divided by the distance between the two setups gives you the taper per inch. Then convert to an angle.

You can also lay the part on a sine plate and dial-indicate the top surface until it reads flat. The shim stack tells you the angle.

What taper is used on machine tool spindles?

Most CNC mills use a 7:24 taper, which is 3.5 inch per foot. Common sizes are CAT 40, CAT 50, BT 30, and BT 40. These are steep and self-releasing, so a drawbar holds the tool.

Older manual mills often use NMTB tapers or R8, and lathes often use Morse tapers in the tailstock.

How do you make a taper on a lathe?

There are three main ways:

Compound rest: swing it to the half angle. Good for short, steep tapers.
Taper attachment: guides the cross slide along a set angle. Good for long tapers and lets you power feed.
Tailstock offset: shift the tailstock sideways. Only works on long, gentle tapers between centers.

How much does a small angle error change a taper?

Error grows with length. A half angle that is off by just 0.1° adds about 0.0035 inch of diameter error per inch of taper length.

Over a 6 inch taper that is about 0.021 inch, which will scrap most fitted parts. On long tapers, get the angle right first and the diameters second.

What is taper percentage?

Taper percent is the diameter change divided by the length, times 100. A 5% taper grows 0.05 units of diameter for every 1 unit of length.

It is the same thing as a ratio, just written another way. A 5% taper equals a 1:20 ratio and 0.6 inch per foot.

Do you measure taper on the diameter or the radius?

Taper rate is always based on the diameter change, not the radius. So 0.600 in./ft. means the diameter grows 0.600 inch over one foot.

Each side only rises half that much. That is why the half angle uses (D − d) ÷ (2 × L) instead of (D − d) ÷ L.

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