Taper Calculator
Enter two diameters and a length — or any two known values — and instantly get taper, taper ratio, taper angle, and taper per foot, with the formula shown for every result.
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Taper Calculator
What Is a Taper Calculator?
A taper calculator converts the raw dimensions of a tapered shaft, hole, or workpiece — its large diameter, small diameter, and length — into every figure a machinist, engineer, or builder actually needs: the taper rate itself, its ratio form, its angle in degrees, and the common shop measurement of taper per foot. Enter what you know, and it solves for what you don't.
Taper describes how gradually a diameter (or width) changes along the length of a part, forming a cone or wedge shape rather than a uniform cylinder. It's central to machining, woodworking, and construction alike — tapered shafts fit into matching tapered sockets with self-locking friction, tapered pins align parts precisely, and tapered structural elements distribute load differently than uniform ones.
The core relationship is simple: Taper = (Large Diameter − Small Diameter) ÷ Length. That single number can then be expressed three different ways depending on the trade and context. As a taper ratio, written "1 : n," it states how many units of length correspond to one unit of diameter change — a very shallow, common machining reference. As a taper angle, calculated using the arctangent function, it states the included angle of the resulting cone in degrees, useful for CAD drawings and angular tolerances. And as taper per foot, a shop-floor convention, it states how much diameter changes over exactly twelve inches of length, regardless of the part's actual length.
This calculator supports five modes to match how the problem comes up in practice. The Taper Calculator mode takes both diameters and the length, and returns taper, ratio, angle, and taper per foot all at once. The Find Large Diameter and Find Small Diameter modes rearrange the same relationship to solve for whichever diameter is unknown, given the other diameter, length, and taper. The Find Length mode solves for the length needed to achieve a specified taper between two known diameters. And the Angle / Ratio Converter mode moves freely between taper angle, taper ratio, taper per unit length, and taper per foot — useful when a drawing specifies one form and a shop needs another.
Every calculation runs live, with the formula shown alongside every result. This tool is built for machinists setting up a lathe for taper turning, engineers specifying tapered fits and fasteners, woodworkers shaping tapered legs or spindles, and construction professionals working with tapered structural or architectural elements — anywhere a precise, repeatable relationship between two diameters and a length needs solving quickly and correctly.
The Complete Guide to Taper Calculations
Everything you need to understand taper, taper ratio, taper angle, and taper per foot across machining and construction.
What Is Taper?
Taper is the gradual, uniform change in the diameter or width of an object along its length, producing a cone or wedge rather than a constant cross-section. It's a fundamental concept in machining, woodworking, plumbing, and structural design, wherever two mating parts need to fit together with a precise, often self-locking, angled connection.
Large and Small Diameter
Every taper calculation starts from two reference diameters: the large diameter (D), measured at the wider end of the tapered section, and the small diameter (d), measured at the narrower end. The difference between them, divided by the length separating them, defines the taper itself.
The Taper Formula
The fundamental formula is Taper = (D − d) ÷ L, giving the rate of diameter change per unit of length. This single value is the foundation every other taper measurement — ratio, angle, and taper per foot — is derived from.
Taper Ratio
Taper ratio expresses the same relationship as "1 : n," meaning the diameter changes by 1 unit for every n units of length. It's calculated as the reciprocal of the taper value, and is a common way taper specifications are written on engineering drawings — a "1:20" taper is far gentler than a "1:5" taper.
Taper Angle
Taper angle expresses the same relationship in degrees, calculated as 2 × arctan(Taper ÷ 2) for the full included angle of the resulting cone, since the taper value describes a diameter change across the full width rather than a single radius. Half of that figure gives the half-angle, measured from the shaft's centerline to its outer surface.
Taper per Foot
Taper per foot (TPF) is a shop-floor convention, particularly common in North American machining, expressing how much the diameter changes over exactly twelve inches of length, regardless of the actual part length: Taper per Foot = Taper × 12. It lets machinists quickly compare tapers of different lengths on a common, standardized basis.
Self-Holding vs. Self-Releasing Tapers
Shallow tapers — generally below roughly 3 degrees included angle — tend to be "self-holding," gripping through friction alone once seated, which is why Morse tapers and similar shallow machine-tool tapers stay put without a locking mechanism. Steeper tapers are "self-releasing," and typically need a separate mechanism (a drawbar, nut, or key) to stay assembled under load.
Common Taper Standards
Certain taper ratios appear repeatedly across machining because they're standardized for interchangeability — Morse tapers, Jacobs tapers, and similar systems used for drill chucks, tool holders, and machine tool spindles all specify precise, published taper ratios that this calculator's formulas apply identically.
Woodworking Applications
Woodworkers use taper calculations for table and chair legs, spindles, and other tapered turnings, where a consistent, calculated taper rate — rather than an eyeballed one — produces a visually balanced, symmetrical result across multiple matching pieces.
Construction and Structural Applications
Tapered structural elements — columns, piles, and certain architectural features — use taper calculations to determine how a cross-section should change along a member's length, balancing material use, load distribution, and aesthetic requirements.
Plumbing and Pipe Fittings
Tapered pipe threads rely on a precisely specified taper rate to create a tightening, self-sealing fit as a fitting is threaded in, distinct from the parallel (non-tapered) threads used elsewhere in plumbing and fastening.
CAD and Engineering Drawings
Engineering drawings specify taper in whichever form suits the application — angle, ratio, or taper per foot — and converting between these forms accurately, without introducing rounding error, is a routine task when moving a design from drawing to shop floor.
Common Taper Calculation Mistakes
The most frequent mistake is confusing taper (a diameter-based rate) with slope or grade (typically a radius- or height-based rate), which use a factor-of-two difference in their angle formulas. Others include mixing units between the two diameters and the length without converting first, confusing the half-angle with the full included angle, using the wrong diameter as "large" versus "small," rounding intermediate results too early across a multi-step conversion, and misapplying a taper-per-foot figure to a part of a very different actual length without first converting back to the underlying taper rate.
Best Practices
Always confirm which taper representation — angle, ratio, taper per foot, or raw taper value — a drawing or specification expects before converting, since mixing conventions produces a part that doesn't match its intended fit. When precision matters, work from the raw taper value (diameter change per unit length) as the common reference point, converting to angle, ratio, or taper per foot only for final reporting or verification.
How the Taper Calculator Works
Four steps take you from raw dimensions to a fully worked taper result.
Enter Known Dimensions
Type in the large diameter, small diameter, and length — or any two known values, depending on the mode.
Choose Calculation Mode
Solve for taper, large diameter, small diameter, length, or convert between angle, ratio, and taper per foot.
Automatic Calculations
The calculator applies the correct taper formula instantly, converting between representations as needed.
Review Results
See taper, ratio, angle, and taper per foot, each labeled clearly alongside the formula used.
Taper Formulas
The core maths behind every taper, ratio, angle, and conversion on this page.
Taper
The rate of diameter change per unit of length.
Taper Ratio
The reciprocal of taper, expressed as "1 : n."
Taper Angle (Full)
The full included angle of the resulting cone, in degrees.
Taper per Foot
Diameter change per 12 inches of length, regardless of actual part length.
Solving for a Diameter or Length
Rearrangements of the core taper formula, used to solve for whichever variable is unknown.
Step-by-Step Taper Examples
Eight worked examples across common machining and construction scenarios.
Example 1 — Basic Taper Calculation
Large diameter 2.5 in, small diameter 1.5 in, length 6 in.
Example 2 — Taper Ratio
Using the taper from Example 1, 0.1667.
Example 3 — Taper Angle
Using the same taper, 0.1667.
Example 4 — Taper per Foot
Using the same taper, 0.1667.
Example 5 — Finding Large Diameter
Small diameter 1 in, length 8 in, taper 0.125.
Example 6 — Finding Small Diameter
Large diameter 3 in, length 10 in, taper 0.1.
Example 7 — Finding Length
Large diameter 2 in, small diameter 1.25 in, taper 0.0625.
Example 8 — Morse Taper No. 2 Reference
A standard Morse Taper No. 2 has a taper of approximately 0.0500 in/in.
Quick Reference Tables
Common taper ratios, angle conversions, and standard machine tapers at a glance.
| Ratio | Taper (in/in) | Full Angle |
|---|---|---|
| 1 : 5 | 0.2000 | 11.42° |
| 1 : 10 | 0.1000 | 5.73° |
| 1 : 20 | 0.0500 | 2.86° |
| 1 : 50 | 0.0200 | 1.15° |
| Taper (in/in) | Taper per Foot |
|---|---|
| 0.0417 | 0.500 in/ft |
| 0.0833 | 1.000 in/ft |
| 0.1667 | 2.000 in/ft |
| 0.2500 | 3.000 in/ft |
| Taper Type | Approx. Taper (in/in) |
|---|---|
| Morse Taper No. 1 | 0.0502 |
| Morse Taper No. 2 | 0.0500 |
| Morse Taper No. 3 | 0.0502 |
| Jacobs Taper No. 2 | 0.0576 |
Benefits of Using a Taper Calculator
Why machinists, engineers, and woodworkers reach for a dedicated tool instead of manual trigonometry.
Instant Calculations
Taper, ratio, angle, and taper per foot appear the moment you enter your dimensions.
Solve for Any Variable
Find large diameter, small diameter, or length from any two known values.
Multiple Taper Formats
Convert freely between taper angle, ratio, taper per unit length, and taper per foot.
Shop-Ready Accuracy
Formulas match standard machining and engineering conventions used on real drawings.
Visual Taper Preview
An SVG illustration shows the tapered shape alongside your calculated results.
Mobile-Friendly Design
All five modes work cleanly on phones and tablets for quick checks on the shop floor.
Practical Applications of the Taper Calculator
Where taper calculations show up across machining, woodworking, and construction.
Lathe taper turning relies on precise taper calculations to set up compound slide angles or taper attachments correctly before cutting a tapered shaft or fitting.
Tool holder and spindle fits — Morse tapers, Jacobs tapers, and similar standardized systems — depend on manufacturers and machinists working from exactly the same taper specification for parts to interchange correctly.
Woodworking uses taper calculations for table and chair legs, spindles, and turned components, where a consistent, calculated taper produces visually matched, symmetrical results across a set of pieces.
Structural and architectural design uses tapered columns, piles, and features where a cross-section needs to change predictably along a member's length.
Plumbing and pipe fitting relies on tapered thread specifications to create a tightening, self-sealing connection as fittings are threaded together.
Precision engineering and CAD drafting use taper conversions constantly when moving a design between angle-based and ratio-based specification conventions.
Tool and die making often requires converting between taper representations to match legacy drawings specified in one convention against modern CNC programming that expects another.
Quality control and inspection use taper calculations to verify that a finished part's measured diameters and length match its specified taper within tolerance.
Common Taper Calculation Mistakes
Avoid these frequent errors when working with taper, ratio, and angle.
Confusing taper with slope
Taper is diameter-based and slope is typically radius- or height-based, differing by a factor of two in their angle formulas.
Mixing units
Diameters and length must be in the same unit before calculating — mixing inches and millimeters produces a meaningless taper value.
Confusing half-angle with full angle
The full included angle is twice the half-angle measured from the shaft's centerline — using the wrong one produces a part that's twice (or half) as steep as intended.
Swapping large and small diameter
Reversing D and d flips the sign of the taper and can produce a nonsensical negative result.
Rounding too early
Rounding intermediate taper values before converting to ratio, angle, or taper per foot compounds error across the final result.
Misapplying taper per foot
Taper per foot describes a rate, not an absolute diameter change — applying it directly to a part of a very different length without converting back to the raw taper rate produces an incorrect diameter.
Using a generic taper for a standardized fit
Morse, Jacobs, and similar standard tapers have precisely published values — using a rounded approximation can produce a part that doesn't seat correctly.
Ignoring measurement tolerance
Small measurement errors in either diameter compound through the taper formula — precise measurement matters more on shallow, long tapers than steep, short ones.
Frequently Asked Questions
Everything you need to know about calculating taper, ratio, and angle.
What is taper in machining and construction?
Taper describes the gradual change in diameter (or width) of a shaft, hole, or workpiece over its length, expressed as a ratio, an angle, or a rate such as taper per foot.
How do you calculate taper?
Subtract the small diameter from the large diameter and divide by the length: Taper = (D − d) ÷ L.
What is the difference between taper and taper angle?
Taper is the rate of diameter change per unit length, while taper angle converts that rate into the included angle of the resulting cone, in degrees.
What is a taper ratio?
A taper ratio, written 1:n, expresses how many units of length correspond to one unit of diameter change, and is the reciprocal of the taper value.
What is taper per foot?
Taper per foot is a shop convention expressing diameter change over exactly 12 inches of length, calculated as Taper × 12.
How do I find a missing diameter or length?
Use the Find Large Diameter, Find Small Diameter, or Find Length modes, entering the two known values plus the taper to solve for the third.
What is a self-holding taper?
A self-holding taper is shallow enough (generally under about 3 degrees included angle) to stay assembled through friction alone, without a separate locking mechanism.
Can I convert between taper angle and taper ratio?
Yes, the Angle / Ratio Converter mode converts freely between taper angle, taper ratio, taper per unit length, and taper per foot.
Does this calculator support standard machine tapers?
You can enter the published taper value for standards like Morse or Jacobs tapers directly to check ratio, angle, and taper per foot for that standard.
Is this calculator accurate?
Yes, it applies the standard taper, ratio, and angle formulas exactly, with configurable decimal precision.
Is it free?
Yes, this Taper Calculator is completely free to use, with no signup or account required.
Does it work on mobile?
Yes. The layout, all five calculation modes, and a sticky mobile CTA are fully responsive across phones and tablets.
Who should use this calculator?
Machinists, engineers, woodworkers, and construction professionals working with tapered shafts, holes, fittings, or structural elements.
What units does this calculator support?
Diameters and length can be entered in inches, millimeters, or centimeters, provided both diameters and the length use the same unit.
Can I copy my results?
Yes, use the Copy Results button to copy your calculation to the clipboard, or Share Results to send it directly from supported devices.
Calculate Taper Instantly
Calculate taper, taper ratio, taper angle, and taper per foot, or solve for a missing diameter or length, with this free, professional Taper Calculator.