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.
Taper Calculator computes every geometry value a machinist or designer needs from three measurements: large diameter (D), small diameter (d), and taper length (L). Enter your dimensions and the tool instantly returns taper angle, included angle, taper rate, taper per foot, taper ratio, slant length, apex distance, full cone length, and diameter change — in both US Customary and Metric units.
What the Taper Calculator Measures
A taper is a uniform reduction in diameter along a straight axis. The calculator derives all angle and rate values from three direct measurements of that geometry.
- D Large Diameter — The wider end of the tapered section, measured in inches (US Customary) or millimetres (Metric). This is the starting diameter before the taper begins to reduce.
- d Small Diameter — The narrower end of the tapered section. Must be strictly less than D. The difference D − d drives every calculated output.
- L Taper Length — The axial distance between the large-diameter and small-diameter measurement planes, measured along the centerline. This is not the slant length of the surface.
Taper angle (also called setup angle) is the angle between the part centerline and one tapered side. It is the half-angle of the cone and the value used to set a compound rest, swivel table, or taper attachment on a lathe. Included angle is the full geometric cone angle from one outer edge to the other — exactly twice the setup angle — and is the format most commonly found on engineering drawings.
Taper per foot (TPF) scales the taper rate to a 12-inch reference length, expressing how many inches the diameter changes per foot of axial travel. TPF is the standard published format in US machining references and is used to identify and match standard tapers — Morse, Brown & Sharpe, Jarno, and others are all defined by their TPF value. Working in Metric, the calculator outputs taper per metre instead.
Taper Calculator Formulas
All results follow from the three inputs using standard right-triangle trigonometry applied to the half-cone cross-section. No material or process factors are applied.
Understanding Each Result
The setup angle (θ) is the half-angle of the cone measured from the centerline to one tapered surface. This is the angle you physically set on the compound rest or taper attachment before cutting. The calculator displays it in decimal degrees and in Degrees-Minutes-Seconds (DMS) format, which matches the graduated scales on most manual lathes and dividing heads, so the value transfers directly to the machine without any conversion.
The included angle (α = 2θ) is the full geometric cone angle spanning both sides of the taper. Engineering drawings and standards — including ASME B5.10 and ISO 1119 — typically specify tapered features by their included angle rather than the half-angle. The calculator shows the included angle in decimal degrees, DMS format, and radians. The radian value is useful when working with trigonometric functions in CAM programming or analytical design calculations.
Taper rate is the dimensionless ratio of diameter change per unit of axial length (in/in or mm/mm). Taper per foot (TPF) scales this ratio to a 12-inch reference — the standard format in US machining tables used to identify Morse, Brown & Sharpe, Jarno, and other named tapers. In Metric mode the output is taper per metre. Slant length (S) is the true edge length along the tapered surface — the hypotenuse of the right triangle formed by the axial length L and the radius change Δr. It differs from L and matters when specifying cutting path length, surface area, or insert engagement length.
Taper ratio (1 : n) expresses how many axial units are needed for one unit of diameter reduction. A ratio of 1 : 6 means the diameter decreases 1 inch for every 6 inches of length — a steeper taper has a smaller second number. Apex distance (A) is the extra axial length from the small-diameter end to the theoretical sharp point where the extended cone would close to zero. Full cone length (L + A) represents the total length of the extrapolated complete cone. These values help verify that the taper geometry is consistent with the part design, fixture envelope, and adjacent features.
Diameter change (ΔD = D − d) is the total reduction from the large to the small end. Radius change (Δr) is half of that — the perpendicular offset from the centerline to the surface at each end. The calculator also shows the equivalent ΔD in the opposite unit system: enter inches and it converts ΔD to millimetres, enter millimetres and it converts to inches. This is useful when working from mixed-standard drawings or verifying a metric part against a US specification without manual arithmetic.
Worked Example
Using the default US Customary values loaded in the calculator:
ΔD = 1.500 − 1.000 = 0.500 in ·
θ = tan⁻¹(0.500 ÷ (2 × 3.000)) = tan⁻¹(0.08333) = 4.764° ·
α = 2 × 4.764° = 9.527° ·
TPF = (0.500 ÷ 3.000) × 12 = 2.000 in/ft ·
Ratio = 3.000 ÷ 0.500 = 6 → 1 : 6.000 ·
S = √(3² + 0.25²) = 3.010 in
How to Read the Results
- 1 Setup angle — Use this value to set the compound rest angle or taper attachment offset on a manual lathe. Rotate the compound to this angle from the centerline before turning. The DMS sub-value can be read directly from the angular scale on most machine tools.
- 2 Included angle — Use this when a drawing or cone specification gives the full cone angle. This equals twice the setup angle. If a print calls out a 10° included angle, the setup angle is 5°.
- 3 Taper per foot — Use this when cross-referencing against US machining standards or taper tables. Match the TPF value to the published value for Morse, Brown & Sharpe, Jarno, or other standard tapers to confirm conformance.
- 4 Taper ratio — Use this to compare the relative steepness of two tapers. A 1 : 3 taper is steeper than a 1 : 10 taper. The ratio also directly states how much axial length is required to achieve one unit of diameter reduction.
- 5 Diameter and radius change — Check these values against the print tolerance before cutting or drawing. Verify that ΔD agrees with the difference between the nominal large and small diameter call-outs, and that Δr is achievable with the tool offset available on the machine.
Assumptions and Limitations
- ▸ All three inputs — D, d, and L — must be positive numbers greater than zero.
- ▸ Large diameter (D) must be strictly greater than small diameter (d). Equal diameters produce no taper; reversed values are geometrically invalid and the calculator will flag the error.
- ▸ The calculator assumes a straight, linear taper — the diameter changes uniformly and proportionally along the full axial length L. Curved tapers, stepped bores, compound tapers, and non-uniform profiles are not modelled.
- ▸ Results are geometric only. The calculator does not account for manufacturing tolerance, tool deflection, surface finish requirements, material springback, chuck or fixture runout, compound slide backlash, thermal expansion, or any process-specific factor.
- ▸ This is a geometry calculator, not a machining-process planner. Feed rate, depth of cut, cutting speed, toolpath strategy, and fixture design must be determined separately based on material, tooling, and machine capability.
- ▸ Inch–millimetre conversion uses the exact international definition: 1 in = 25.4 mm (NIST SP 330), applied without rounding to the raw diameter-change value before display.
References
- Machinery's Handbook (Industrial Press, current edition) — Primary reference for taper definitions, taper per foot calculations, standard taper series (Morse, Brown & Sharpe, Jarno, and others), and machining geometry conventions used throughout this tool.
- ASME B5.10 — Machine Tapers: Self-Holding and Steep Taper Series (American Society of Mechanical Engineers) — Dimensional and angular standards for machine tool tapers, defining taper per foot, cone angle, and gage line references for US practice.
- ISO 1119:1998 — Series of Conical Tapers and Taper Angles (International Organization for Standardization) — Defines the preferred international series of taper ratios and included angles used in engineering drawings and metric machining specifications.
- NIST Special Publication 330 (National Institute of Standards and Technology, current edition) — The International System of Units (SI): source for the exact inch–millimetre conversion factor (1 in = 25.4 mm) applied to the equivalent-unit diameter change output.
- General machining geometry — Setup angle, included angle, slant length, apex distance, and full cone length derivations follow standard right-triangle trigonometry applied to the half-cone cross-section, consistent with methods described in engineering technology references covering turning and boring operations.