Taper Calculator

Taper Calculator for Machined and Tapered Parts

Use this taper calculator to work out the angle, end diameter, or length of a straight tapered part. Enter the large diameter, small diameter, and taper length to calculate the half angle, included angle, taper ratio, and taper per foot or meter. Or choose a solve mode to find a missing diameter or length from an angle you already know.

A taper is a steady change in diameter over a length. It appears on shafts, pins, tool holders, machine centers, pipe features, furniture legs, and many turned components. The calculator supports metric and imperial units, lets you choose whether the entered angle is a half angle or included angle, and can find the diameter at an optional position along the taper.

The results describe basic straight-taper geometry. They do not identify a standard machine taper, check fit, calculate load capacity, account for a curved profile, or replace a drawing tolerance. If the part must mate with a gauge, socket, tool holder, or other component, use the specified standard and inspection method for that fit.

What Is a Taper?

A taper is a gradual, usually straight-sided change in diameter along an axis. One end is larger than the other. The difference between the two end diameters is called the diameter change or taper difference. Divide that difference by the axial length to describe how quickly the diameter changes per unit length.

Taper is commonly written as a ratio such as 1:20. In this convention, the diameter changes by one unit for every twenty units of axial length. Machinists may also describe taper per foot or taper per inch. These are diameter changes over a stated reference length, not the change in radius on just one side.

The angle of one tapered side measured from the centerline is the half angle. The angle between the two opposite tapered sides, if the part is extended through its axis, is the included angle. The included angle is twice the half angle for a symmetric straight taper. Lathe compound settings commonly use the half angle, while drawings may call out either an angle or a taper ratio, so read the dimension carefully.

Taper Angle Formula

Let D be the large diameter, d be the small diameter, and L be the axial length of the tapered section. The half angle α is calculated from the right triangle made by the radius change and the axial length: α = arctan((D − d) ÷ (2L)). The diameter difference is divided by two because each side moves inward by half that amount from the centerline.

The included angle is 2α. For example, if a part reduces from 25 mm to 20 mm across a 100 mm taper length, the diameter change is 5 mm. The radial change is 2.5 mm, so the half angle is arctan(2.5/100), about 1.432 degrees. The included angle is about 2.864 degrees.

When using a known half angle, find the small diameter with d = D − 2L × tan(α). To find the taper length from both end diameters and a known half angle, rearrange the equation: L = (D − d) ÷ (2 × tan(α)). If the angle you entered is an included angle, divide it by two before applying the half-angle relationship.

These formulas assume a straight taper with circular cross-sections and a constant slope. A curved, compound, stepped, or blended profile needs a different model. Keep diameter and length in the same unit before calculating, although the angle itself is independent of whether those consistent dimensions are inches or millimeters.

Taper Ratio, Taper per Foot, and Taper per Meter

Taper ratio describes the diameter change compared with the axial length. With ΔD = D − d, a ratio of 1:n has n = L ÷ ΔD. For example, a 5 mm diameter change over 100 mm gives a 1:20 taper. The ratio is often useful for comparing similar parts that have different overall sizes.

Taper per foot is common in inch-based machining. It is calculated as diameter change divided by taper length, multiplied by 12 when the length is entered in inches. If a part changes 0.25 inch in diameter over 6 inches, its taper per foot is 0.25 ÷ 6 × 12, or 0.5 inch per foot. This is the diameter change extrapolated to a 12-inch length at the same slope.

In metric work, it may be useful to express taper per meter: multiply the diameter change per millimeter by 1,000. A 5 mm diameter change over 100 mm corresponds to 50 mm per meter. The value is a normalized rate; the actual part only changes by the amount over its own taper length.

Do not confuse taper per foot with the half angle. Taper per foot uses the full diameter change, while the half-angle triangle uses half of that change as radial rise. Both describe the same geometry when their units and definitions are applied correctly.

Lathe Compound Rest and Taper Turning

For a simple straight taper turned on a lathe, the compound rest is commonly set to the half angle from the workpiece centerline. If the drawing specifies an included angle, halve it before entering a compound-rest angle. Always confirm the machine's scale convention because some equipment markings or setup notes may describe angles differently.

A small angular error can cause a noticeable diameter mismatch over a long taper. The calculator shows the decimal degree value and the diameter change over the entered length. For precision work, verify the taper using the required gauge, sine method, indicator setup, or other inspection procedure specified for the part.

The tailstock-offset method is another way to turn some long, shallow tapers between centers. Its setup depends on the work length, center arrangement, and taper dimensions. This calculator gives taper geometry but does not calculate machine offsets, tool clearances, center travel, or workholding requirements. Refer to the lathe manual or a machining reference for the chosen setup method.

Tool wear, deflection, material hardness, setup rigidity, and finishing passes can affect the produced angle and end diameters. Treat the calculated angle as a setup target, then measure the finished part and adjust according to the process plan and tolerances.

Finding Diameter at a Position Along the Taper

A straight taper changes diameter at a constant rate. If x is the distance from the large end, the diameter at that position is D(x) = D − ((D − d) × x ÷ L). This assumes the taper begins at the large end and runs evenly to the small end across the entered length.

For example, halfway along a straight taper, the diameter is halfway between the large and small end diameters. This station value can help with a preliminary measurement location, stock allowance, or a drawing check. It does not account for a lead-in, relief, shoulder, runout, or blended section outside the defined taper length.

Enter the distance from the large end in the same length unit as the taper. Keep the station between zero and the taper length. At zero, the calculated diameter is the large end diameter; at the full length, it is the small end diameter. A station beyond the section would mathematically extend the straight taper, but this calculator keeps the optional check within the entered section.

How to Use This Taper Calculator

Choose metric or imperial units, then select whether to find the angle, small diameter, or taper length. For an angle calculation, enter both end diameters and the taper length. To find the small diameter, enter the large diameter, taper length, and angle. To find length, enter both diameters and the angle. Select half angle or included angle to match the value on your drawing.

Check that the large diameter is greater than the small diameter and that all required dimensions are positive. If you want a diameter at a point along the taper, enter the station distance from the large end. Read the calculated taper ratio and taper per foot or meter along with the angle and solved dimension.

Use the formula steps to review each substitution. You can copy the result, save it in your browser, export a CSV file, or print the calculation for a shop note. Confirm that the drawing uses the same taper length definition and angle convention before relying on a number for setup or inspection.

Machine Tapers and Common Calculation Mistakes

A generic taper calculation does not identify a standard machine taper such as Morse, Jarno, Brown & Sharpe, or a pipe thread taper. Standard tapers have defined dimensions, gauge planes, tolerances, and fit requirements. If a part must mate with a standard socket or gauge, identify the exact standard and size from the drawing or equipment documentation.

One common error is using the included angle where a lathe setup expects the half angle. Another is dividing the diameter change by the full length when calculating the angle; the half-angle formula needs the radius change, so it divides by twice the length. Taper per foot, however, uses the full diameter change over the length.

Other mistakes include mixing metric and inch units, measuring the overall part length instead of the tapered section length, reversing the large and small ends, and ignoring a shoulder or relief. A good result depends on accurate dimensions and a clear definition of where the taper starts and ends. For a precision fit, use the specified standard and verify the finished geometry with suitable inspection equipment.

Taper Fit, Tolerances, and Inspection

Two parts can share the same nominal taper angle and still fail to fit if their diameters, gauge lengths, surface finish, or tolerances do not match. A machine taper may be checked at a specified gauge plane rather than at the physical end of the part. The reference location and permitted deviation should come from the drawing or the standard that defines the mating parts.

For a general tapered shaft, the end diameters and length describe the nominal geometry. A production drawing may also include diameter tolerances, angular limits, straightness, runout, or a surface finish requirement. Those details affect how a part is made and inspected. The calculator solves the geometric relationship but does not infer missing tolerances or determine whether an assembly will hold under load.

Measurement methods depend on the part size, taper length, and accuracy required. A coordinate measuring machine, taper gauge, dial indicator setup, sine bar, or contact measurement may be used in different shops. Use a method suited to the drawing and avoid relying on an angle reading alone when the fit is controlled by a gauge diameter or functional contact.

Planning Stock and Checking a Tapered Part

Before cutting, compare the raw stock diameter with the required large and small diameters. Make sure the taper begins and ends where the drawing indicates, and allow for any shoulder, relief, lead-in, or finishing stock outside the tapered section. The length entered in the calculator should describe only the sloped section unless the drawing defines it another way.

After machining, measure both end diameters at the specified locations and verify the taper length. If the end diameters are correct but the middle is not, the surface may not be straight or the taper may have a barrel or hollow shape. A station diameter calculation gives the ideal linear value for comparison, but intermediate measurements still need an appropriate tool and inspection plan.

Record the angle convention with the setup. Writing 'half angle from centerline' or 'included angle between sides' beside a decimal degree value helps prevent a later operator from doubling or halving the setting by mistake. Clear notes are especially useful when a setup passes between design, programming, machining, and inspection teams.