The Geometry of Truncated Cones
A machined taper is geometrically a truncated cone, or frustum. Its shape is defined by four interdependent dimensions: the large diameter, the small diameter, the taper length, and the half angle. The half angle, denoted as α, is the angle between the central axis of the cone and its sloped outer wall. The included angle is the total angle from one sloped side to the other, which is exactly twice the half angle (2α).
The rate at which the diameter changes relative to the length is the taper rate, represented by the variable k. This rate is calculated using the formula:
k = (D - d) / L
Where D is the large diameter, d is the small diameter, and L is the taper length. The half angle is mathematically tied to this rate through the tangent function:
α = arctan(k / 2)
Because these geometric properties are bound by exact mathematical identities, knowing any three of these dimensions allows you to solve for the fourth. The Taper Angle Calculator automates these calculations, allowing you to select your target variable using the Solve for setting. You can choose between Taper & angle, Taper length, Small diameter, or Large diameter to compute the missing dimension.
Lathe Setup and the Compound Rest
When turning a taper on a manual lathe, the machinist must set the compound rest or the taper attachment to the correct angle. A common and costly mistake is setting the machine to the total included angle rather than the half angle.
Because the workpiece rotates around its central longitudinal axis, any cut made by the tool bit is mirrored on the opposite side of the part. The cutting tool travels along a single sloped path parallel to the compound rest. Therefore, the compound rest must always be set to the half angle. Setting the compound rest to the included angle by mistake doubles the intended taper rate, reducing the workpiece to scrap.
For example, a Morse 2 taper has a nominal included angle of approximately 2°52′. To turn this taper, the lathe operator must set the compound rest to the half angle, which is approximately 1°26′.
Taper Measurement Conventions
Different industries and regional standards express taper rates in distinct ways. The Taper Angle Calculator computes and displays these values in three primary formats under the Taper geometry heading:
- Taper rate (diameter change per unit length): This is the raw decimal value k, representing the change in diameter per single unit of length.
- Taper per foot: Expressed in inches per foot (in/ft), this is the traditional American machining standard. It is calculated by multiplying the taper rate by 12 (12 × k).
- Taper ratio: Expressed in the format 1:X, where X is the ratio denominator (X = 1 ÷ k). This format is widely used on metric drawings and ISO standards to show that the diameter changes by 1 unit over a length of X units.
The tool also provides the Diameter change per meter (1000 × k) in millimetres per meter (mm/m) when working with metric dimensions.
Self-Holding vs. Self-Releasing Tapers
Machine tapers generally fall into two functional categories based on their angle: self-holding tapers and self-releasing (or steep) tapers.
Self-Holding Tapers
Self-holding tapers feature very small angles, typically under 3°. The friction between the male and female mating surfaces is strong enough to wedge the components together, allowing them to drive cutting tools without slipping, even without a drawbar. Common examples include Morse, Brown & Sharpe, and Jarno tapers. Because the angles are so shallow, these tapers require a drift key or drawbar force to separate.
Self-Releasing Tapers
Self-releasing tapers, such as the 7:24 steep taper used in CAT, BT, and ISO toolholders, have much larger angles. The nominal taper rate is 3.5 inches per foot, which corresponds to an included angle of 16°35′39″. These tapers do not stick or bind in the spindle. They rely entirely on a drawbar to hold them in place and release instantly when the drawbar tension is relieved.
Standard Taper Families
The calculator compares your calculated taper rate against several historical and modern standard taper families. If the calculated rate is within a 1% relative deviation of a standard, the tool displays the match under Closest standard taper.
| Taper Standard | Nominal Taper Rate (Taper per Foot / Ratio) |
|---|---|
| Morse MT0 | 0.62460 in/ft |
| Morse MT1 | 0.59858 in/ft |
| Morse MT2 | 0.59941 in/ft |
| Morse MT3 | 0.60235 in/ft |
| Morse MT4 | 0.62326 in/ft |
| Morse MT5 | 0.63151 in/ft |
| Morse MT6 | 0.62565 in/ft |
| Brown & Sharpe | 0.50000 in/ft (1:24) |
| Jarno / Metric 80–200 | 0.60000 in/ft (1:20) |
| Metric 4/6 (ISO 296) | 1:19.254 |
| NPT Pipe Thread | 0.75000 in/ft (1:16) |
| 7:24 Steep Taper | 3.50000 in/ft |
Several self-holding standards cluster very close to one another. For example, a Morse 2 taper (0.59941 in/ft) and a Jarno taper (0.60000 in/ft) differ by only about 0.098%. Because these standards sit within the 1% matching tolerance, the calculator's match is a helpful hint rather than a definitive identification. Physical verification with standard gauges is always required.
Metrology and Taper Verification
While the calculator provides mathematically perfect outputs, physical parts are subject to machining tolerances and measurement errors. To verify a physical taper, machinists use several high-precision metrology tools:
- Sine Bars and Gauge Blocks: By placing the tapered part on a sine bar and adjusting gauge blocks until a dial indicator shows zero deviation along the top edge, you can verify the half angle to arc-second precision.
- Taper Plug and Ring Gauges: These precision-ground reference tools are coated with layout fluid (such as Prussian blue) and inserted into or onto the workpiece. A twisting motion reveals contact high spots, showing whether the taper angle matches the standard.
- Micrometer Measurements: Measuring the diameter at two known distances along the taper allows you to calculate the physical taper rate to compare against the theoretical values generated by the tool.
Calculation Rules and Input Limits
To ensure physical and mathematical validity, the calculator enforces several input rules and constraints:
- Numeric Validation: All inputs must be valid numbers. If an invalid character is entered, the tool displays:
‹field›: “‹token›” is not a number.. - Positive Values: Every input must be greater than zero. If zero or a negative number is entered, the tool displays:
‹field› must be greater than zero.. - Sanity Cap: To prevent extreme values, inputs are capped at 1,000,000. Exceeding this displays:
‹field› must be 1,000,000 or less — a sanity cap, not a machining limit.. - Diameter Order: The small diameter must be strictly less than the large diameter. If it is not, the tool displays:
The small diameter must be less than the large diameter — otherwise the section is a cylinder or flares the wrong way.. - Angle Range: The half angle must be greater than 0° and less than 90°. Violating this displays:
The half angle must be greater than 0° and less than 90°.. - Centerline Intersection: If a combination of length and angle causes the taper to cross the centerline, the small diameter would mathematically become negative. In this case, the tool displays:
That length and angle would taper past the centerline — the small diameter comes out negative. Shorten the length or reduce the angle.. - Privacy: All calculations are processed locally in your browser. Your dimensions and results are never uploaded or sent to an external server.
Frequently Asked Questions
Do I set the lathe compound rest to the half angle or the included angle?
The half angle. The included angle is measured across both sloped sides of the taper, but the cutting tool travels along just one side while the part spins around the centerline, so the compound rest — and any taper attachment — is set to the angle between one side and the axis. A Morse 2 taper has an included angle of about 2°52′, so the compound is set to roughly 1°26′. Setting the included angle by mistake doubles the taper and scraps the fit.
What is the difference between taper per foot, a 1:X ratio and the taper rate?
All three state the same geometry — how quickly the diameter shrinks along the length. The taper rate k = (D - d) ÷ L is the diameter change per unit length: a rate of 0.05 means the diameter drops 0.05 mm for every millimetre of taper. Taper per foot multiplies that by 12 — a rate of 0.05 is 0.6 in/ft — the American shop convention. A 1:X ratio inverts it: X = 1 ÷ k, so a rate of 0.05 is a 1:20 taper. That is the convention on ISO drawings and for Morse and metric tapers. This calculator shows all three.
How do I identify which standard taper a part or socket has?
Measure the diameter at two points along the taper with a micrometer, measure the distance between those points, and compute the taper rate — that is exactly what this calculator does. Compare the rate with the standard values: Morse tapers run from about 0.599 to 0.632 in/ft, Brown & Sharpe is 0.5, Jarno and the larger metric tapers are 1:20, and NPT pipe threads are 1:16. Because several self-holding tapers sit within 1% of each other, a measured rate can only narrow the field — confirm the identification with plug or ring gauges, or by test-fitting the mating part, before machining anything to match.
How precise is the result — is there a safety factor?
The math is exact: for a perfect cone these formulas are identities, so there is no safety factor to apply. The uncertainty lives in your measurements and your machine, not in the arithmetic — rounding the half angle to 0.001° shifts the small end by only about 0.0035 mm per 100 mm of taper length, far below typical micrometer error. What matters is verifying the cut taper: a sine bar with gauge blocks and a dial indicator reaches arc-second accuracy, and plug or ring gauges tell you directly whether the taper fits.