Taper Calculator (taper angle, ratio, taper per foot, tailstock set-over)

Understanding the taper calculator — concept and principle

A taper is a cone-shaped section whose diameter changes steadily along its length, as on lathe centres, drill shanks and tool holders. This calculator turns the large diameter, small diameter and length into the included angle, the half angle you set on a compound rest, the taper ratio, taper per foot and percent, and, if you give the work length, the tailstock set-over.

The diameter drops 10 mm over 50 mm: a 1 : 5 taper with a 5.71° half angle.

How it is calculated

Taper per unit length = (D − d) ÷ L

Half angle = atan((D − d) ÷ 2L); included angle = 2 × half angle

Taper ratio = 1 : L ÷ (D − d)

Taper per foot = 12 × (D − d) ÷ L; taper (%) = 100 × (D − d) ÷ L

Tailstock set-over = (D − d) × W ÷ (2L)

D large diameter, d small diameter, L taper length, W total work length between centres; all in the same unit.

The principle

Cut a taper lengthwise and you see a trapezoid. On each side, the radius drops by (D − d)/2 over the length L, so the tangent of the half angle is (D − d)/(2L). A lathe compound rest is swung to that half angle because the tool moves along one side of the cone. Taper per foot and ratio are just other ways of writing the same slope: inch standards quote the diameter change over 12 inches, while metric drawings often give 1 : x. With the tailstock offset method, the whole part between centres is tilted, so the offset is the radius change scaled up from the taper length to the full length W.

Worked example

Default inputs: D = 30 mm, d = 20 mm, L = 50 mm, W = 0 (set-over skipped).

D − d = 10 mm; taper per unit length = 10 ÷ 50 = 0.2.
Half angle = atan(10 ÷ 100) = atan(0.1) = 5.7106°.
Included angle = 2 × 5.7106 = 11.4212°.
Taper ratio = 1 : 50 ÷ 10 = 1 : 5.
Taper per foot = 12 × 0.2 = 2.4 in/ft; taper = 20 %.
With W = 0, the tailstock set-over is not computed (—).

Each point is the calculator re-run with only “Length of the taper L (mm)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Length of the taper L (mm)	Included angle (°)
12.5	43.6028
25	22.6199
37.5	15.1893
50	11.4212
62.5	9.1478
75	7.6281
100	5.7248

How to read the result

Small included angles (a few degrees) give self-holding tapers that lock by friction, as on Morse tapers. Steeper tapers release easily and are used where parts must come apart freely. Use the half angle for the compound-rest or taper attachment setting and the included angle when comparing with drawings. Check the result against a gauge or by blueing, because a slight angle error changes the fit noticeably.

Common mistakes

Setting the compound rest to the included angle instead of the half angle.
Using different units for diameters and length.
Using the taper length for set-over when the work between centres is longer.

Keep in mind: Tailstock set-over is an approximation and can strain the centres on steep tapers. For precision work, check standard taper tables and verify with gauges.

Frequently asked questions

What is the difference between taper ratio and taper per foot?

Both describe the same slope. The ratio gives length per unit of diameter change; taper per foot gives diameter change over 12 inches.

Which angle do I set on the compound rest?

The half angle, because the tool follows one side of the cone.

Can I use inches instead of millimetres?

Yes, as long as all lengths use the same unit; taper per foot is always in inches per foot.

Sources: Standard taper formulas: (D − d)/L, half angle atan((D − d)/2L), taper per foot and tailstock set-over (D − d)·W/(2L)

Checked 9 Nov 2026. Taper (D − d)/L, half angle atan((D − d)/2L), taper per foot and tailstock set-over (D − d)·W/(2L): standard formula.

Your numbers