Taper Calculator uses D, , and to calculate setup angle, included angle, taper rate, taper per foot, taper ratio, slant length, and diameter change for machining taper geometry.
The number machinists dial into the compound rest is not the included angle of the taper — it is half of it. Confusing the two is the single most common setup error when turning a taper on a lathe, and the tool above makes the distinction explicit: the large Setup Angle / Taper Angle displayed in the hero is the half-angle you actually rotate the compound, while Card 1 shows the full included angle that describes the cone geometry. Both matter; they are used in different places on the machine and in the drawing.
The calculator accepts any combination of large diameter, small diameter, and taper length — in either inches or millimeters — and returns every value a machinist, toolmaker, or inspector needs: setup angle, included angle, taper per foot (or taper per meter in metric), taper ratio, slant length, apex distance, and full cone length.
Formulas
All outputs are derived from three inputs: large diameter D, small diameter d, and axial taper length L. The variable names below match the labels in the calculator.
Diameter and Radius Change
- Diameter Change (ΔD) = D − d
- Radius Change (Δr) = ΔD ÷ 2
Taper Angle (Setup / Half-Angle)
- Half-Angle (radians) = arctan( ΔD ÷ (2 × L) )
- Half-Angle (degrees) = Half-Angle (radians) × (180 ÷ π)
Included Angle
- Included Angle (degrees) = Half-Angle (degrees) × 2
- Included Angle (radians) = Included Angle (degrees) × (π ÷ 180)
Degrees–Minutes–Seconds Conversion
- Degrees (whole) = floor( decimal degrees )
- Minutes = floor( (decimal degrees − whole degrees) × 60 )
- Seconds = round( (minutes remainder) × 60 )
Taper Rate
- Taper per unit (in/in or mm/mm) = ΔD ÷ L
- Taper per Foot (US) = (ΔD ÷ L) × 12
- Taper per Meter (metric) = (ΔD ÷ L) × 1000
Slant Length
- Slant Length = √( L² + Δr² )
Taper Ratio (1 : N)
- N = L ÷ ΔD
- Expressed as: 1 : N (N is the axial length per unit of diameter reduction)
Apex Distance and Full Cone Length
- Apex Distance = (d × L) ÷ ΔD
- Full Cone Length = L + Apex Distance
Cross-Unit Diameter Conversion
- ΔD in mm (US mode) = ΔD (inches) × 25.4
- ΔD in inches (metric mode) = ΔD (mm) ÷ 25.4
How It Works
The calculator starts from the right triangle formed by the taper. The two legs are the axial length L (horizontal) and the radius change Δr = (D − d) ÷ 2 (vertical). The arctangent of the ratio of those two legs gives the half-angle — the angle at the large-diameter end of that triangle. That is the value displayed in the hero as the Taper Angle / Setup Angle, expressed in decimal degrees with DMS below it.
Doubling the half-angle gives the Included Angle shown on Card 1. This is the full cone angle that would appear on a technical drawing’s dimension. The same angle is also reported in radians for CAM software or trigonometric reference.
Card 2 reports the Taper Rate: the raw ratio ΔD ÷ L gives inches-per-inch (or mm/mm), which is multiplied by 12 in US mode to yield Taper per Foot — the traditional TPF value printed in most American machining handbooks and engraved on lathe taper-attachment scales. In metric mode the same ratio is multiplied by 1000 for Taper per Meter. Card 2 also shows the Slant Length, the true hypotenuse edge of the cone’s surface, computed via the Pythagorean theorem using L and Δr.
Card 3 reports the Taper Ratio as 1 : N, where N is the axial length required for the diameter to decrease by exactly one unit. It also calculates the Apex Distance — the theoretical distance you would need to extend the small-diameter end before the cone narrows to a point — and the Full Cone Length (taper length plus apex distance), which is useful when machining a complete truncated cone from bar stock.
Card 4 isolates the Diameter Change and Radius Change as standalone numbers. It also converts the diameter difference to the alternate unit system (mm if you entered inches, or inches if you entered mm) so the value can be verified against a drawing dimensioned in the other system without a separate calculation.
Switching the Measurement System dropdown resets the inputs to sensible metric or US defaults and recalculates immediately. All other inputs recalculate live as you type — the Calculate button scrolls to the results but is not required to trigger the math.
The Apex Distance Is a Theoretical Value, Not a Physical One
Card 3’s Apex Distance and Full Cone Length are calculated by projecting the taper’s walls inward until they converge at a point. The formula is (d × L) ÷ ΔD. This produces a real, positive number for any valid input — but that point almost never physically exists on the workpiece. The taper you are machining is a truncated cone: it has two flat ends, not a tip. The apex is a geometric construction used to verify that the taper angle is consistent and to locate the cone in space for tolerance analysis.
Where this matters in practice: if you are checking a taper plug against a ring gauge and the fit is correct at the large end but loose at the small end, comparing the measured apex distance to the calculated one can reveal whether the diameter change is correct but the length is short, or vice versa. The two quantities are not interchangeable, but the apex distance ties them together in a way that a standalone angle measurement does not.
Worked Example: Morse Taper No. 2 Shank
A toolmaker is verifying the taper on a replacement Morse Taper No. 2 arbor. The print calls for a large diameter of 0.700 in, a small diameter of 0.572 in, and a taper length of 2.560 in. Entering these values in US Customary mode and clicking Calculate returns the following:
- Taper Angle / Setup Angle (hero): 1.432° — DMS shown as 1° 25′ 54″. This is the compound rest angle.
- Card 1 — Included Angle: 2.864°, or 2° 51′ 50″ in DMS. This is the full cone angle as it would appear on the drawing.
- Card 2 — Taper Rate: 0.0500 in/in. Taper per Foot reads 0.600 in/ft. Slant Length is 2.561 in — barely longer than the axial length because the taper is shallow.
- Card 3 — Taper Ratio: 1 : 20.000. Apex Distance is 11.429 in, meaning the cone’s theoretical tip is 11.4 inches beyond the small end of the taper. Full Cone Length is 13.989 in.
- Card 4 — Diameter Change: 0.128 in total, 0.064 in radius change. Equivalent metric ΔD is 3.251 mm.
The 1 : 20 ratio matches the Morse No. 2 standard nominal ratio, confirming the geometry is correct. The toolmaker notes the TPF of 0.600 in/ft as the value to set on the lathe’s taper attachment.
Frequently Asked Questions
Why does the calculator refuse to compute when D equals d?
The validation requires the large diameter to be strictly greater than the small diameter (D > d). Equal diameters produce a ΔD of zero, which causes a division-by-zero in the taper ratio and arctangent formulas. A zero-taper cylinder is not a taper at all, so equal values are rejected and a geometry-check warning appears.
What is the difference between the Setup Angle and the Included Angle?
The Setup Angle (shown in the hero) is the half-angle — the angle between the taper’s axis and one sloped surface. It is the value you rotate the compound rest or taper attachment on a lathe. The Included Angle on Card 1 is twice that value, spanning both sides of the cone. Drawings and standards typically call out the included angle; machines are set to the half-angle.
When I switch between US and Metric, the inputs change. Is my previous entry lost?
Yes. Switching the Measurement System dropdown resets all three inputs to built-in defaults (1.500 / 1.000 / 3.000 in for US; 40.0 / 30.0 / 50.0 mm for metric) and recalculates. The reset happens because the input values are in different units and a direct substitution without conversion would produce a meaningless result. Convert your dimensions to the target unit first, then enter them after switching.
Card 2 shows Taper per Foot in US mode. How is that calculated?
Taper per Foot (TPF) = (ΔD ÷ L) × 12. The calculator first finds the taper per inch, then multiplies by 12 to scale to a 12-inch reference length. In metric mode the same base ratio is multiplied by 1000 to give taper per meter (mm/m).
What does the Slant Length on Card 2 actually represent?
Slant Length is the true edge length along the cone’s outer surface — the hypotenuse of the right triangle whose legs are the axial length L and the radius change Δr. It is longer than L whenever the taper has any angle at all, and is the dimension you would measure with a surface-plate height gauge along the sloped face, not along the bore axis. For shallow tapers the difference from L is small; for steep tapers it can be significant.
The DMS readout sometimes shows 60 seconds or 60 minutes briefly — is that a bug?
The code handles this explicitly: if rounding produces 60 seconds, seconds reset to 0 and minutes increment by 1; if that produces 60 minutes, minutes reset to 0 and degrees increment by 1. The carry-propagation runs before the value is displayed, so you should not see 60 appear in a final result. If it appears briefly while you are still typing an input, it will self-correct as soon as the field resolves to a stable value.