Quick answer
Taper = (D − d) ÷ L, and the half angle = arctan((D − d) ÷ 2L). A taper from 2.0 to 1.5 in over 6 in is 0.08333 per unit, 1 per 12 units of length, with an included angle of 4.7719° and a half angle of 2.3859°.
Enter the large and small diameters and the taper length. The calculator returns the taper, taper per 12 and the angles.
Related tools: Square Footage Calculator: Area of a rectangular space in square feet, square metres and square yards. Concrete Calculator: Cubic yards of concrete for a slab, plus bag counts.
How to use the Taper Calculator
- Enter large diameter (above 0).
- Enter small diameter (above 0).
- Enter length of taper (above 0).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Large diameter | A number | > 0 |
| Small diameter | A number | > 0 |
| Length of taper | A number | > 0 |
Formula and method
Formula
taper = (D − d) ÷ L; half angle = arctan((D − d) ÷ 2L)
IEEE-754 double precision; shown to at most 6 decimal places.
Use one unit for all three; taper per 12 units is the usual 'taper per foot' when lengths are in inches.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Large diameter = 2.0; Small diameter = 1.5; Length of taper = 6.
| Result | Value |
|---|---|
| Taper | 0.083333 |
| Taper per 12 units of length | 1 |
| Included angle | 4.7719° |
| Half angle | 2.3859° |
Example 2
Inputs: Large diameter = 50; Small diameter = 40; Length of taper = 100.
| Result | Value |
|---|---|
| Taper | 0.1 |
| Taper per 12 units of length | 1.2 |
| Included angle | 5.7248° |
| Half angle | 2.8624° |
Example 3
Inputs: Large diameter = 1.25; Small diameter = 0.75; Length of taper = 4.
| Result | Value |
|---|---|
| Taper | 0.125 |
| Taper per 12 units of length | 1.5 |
| Included angle | 7.1527° |
| Half angle | 3.5763° |
The first line is the headline result (taper). Below it the taper Calculator lists taper per 12 units of length, included angle and half angle, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Reading it
Taper per foot is the change in diameter over 12 units: here 1 inch per foot. The included angle is twice the half angle, which is how a lathe's compound rest is set (half angle).
Machining and woodworking
Standard tapers such as Morse and Jacobs are defined this way. The large diameter must exceed the small one. Use consistent units for all three inputs.
How the result changes with large diameter
Holding every other input at the first worked example, this table changes only large diameter. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Large diameter | Taper | Taper per 12 units of length | Included angle |
|---|---|---|---|
| 2 | 0.083333 | 1 | 4.7719° |
| 4 | 0.416667 | 5 | 23.5366° |
| 8 | 1.083333 | 13 | 56.8859° |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Small diameter: entering "2" shows "The small diameter must be less than the large diameter." beside that field and no result.
- Length of taper: entering "0" shows "Length of taper must be greater than 0." beside that field and no result.
Choosing between the Taper Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Square Footage Calculator](/construction/square-footage-calculator/) | length, width and unit | Area |
| [Concrete Calculator](/construction/concrete-calculator/) | length, width and depth / thickness | Cubic yards |
| [Stair Calculator](/construction/stair-calculator/) | total rise, preferred riser height and tread depth | Number of risers |
| [Roofing Calculator](/construction/roofing-calculator/) | building length, building width and roof pitch | Roof area |
| [Gravel Calculator](/construction/gravel-calculator/) | length, width and depth / thickness | Cubic yards |
Background: construction estimates
Material quantities are geometry plus a margin. Calculate the exact volume or area, add an allowance for waste, spillage, cuts and uneven ground (calculator pages commonly mention 5 to 10 percent, a rule of thumb that varies by material and job) and round up to whole units. One cubic yard is exactly 27 cubic feet.
Suppliers often have minimum order sizes and may charge for small loads, so confirm quantities, product yields and local code with your supplier or a qualified estimator before ordering.
Assumptions and limitations
- Large diameter must exceed small diameter
Quantities are estimates. Add allowances for waste and site conditions, and follow local codes and a qualified professional's instructions for structural or safety-critical work.
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for small diameter and length of taper to confirm this.
Frequently asked questions
What is a Morse taper?
A standard taper used in machine tools, about 5/8 inch per foot.
Is taper per foot a ratio?
Yes, change in diameter per 12 inches.
What is the half angle used for?
To set a lathe or mill.
What does the Taper Calculator calculate?
Taper ratio and angle from large diameter, small diameter and length. It returns Taper, Taper per 12 units of length, Included angle and Half angle.
What formula does the Taper Calculator use?
taper = (D − d) ÷ L; half angle = arctan((D − d) ÷ 2L). Use one unit for all three; taper per 12 units is the usual 'taper per foot' when lengths are in inches.
What do I need to enter in the taper Calculator?
You enter Large diameter, Small diameter and Length of taper.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Large diameter must exceed small diameter.
What is an example of a taper result?
With large diameter 2.0, small diameter 1.5 and length of taper 6, the taper is 0.083333.
Where does the method come from?
The method follows: Machinery's Handbook, taper formulas.
Can I rely on this calculator for a decision?
Quantities are estimates. Add allowances for waste and site conditions, and follow local codes and a qualified professional's instructions for structural or safety-critical work. For this calculator in particular: Large diameter must exceed small diameter.
Related calculators
- Square Footage Calculator — Area of a rectangular space in square feet, square metres and square yards.
- Concrete Calculator — Cubic yards of concrete for a slab, plus bag counts.
- Roofing Calculator — Roof surface area and roofing squares from the footprint and pitch.
- Gravel Calculator — Cubic yards of gravel for an area.
- Roof Pitch Calculator — Roof pitch as rise per 12 inches, angle, slope percent and rafter length factor.
Sources
- Machinery's Handbook, taper formulas
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.