Taper calculator
A part that goes from 1 inch to 0.5 inch over 4 inches has a taper of 0.125 per inch and a half-angle of a little over three and a half degrees. Measure both ends and the length, then set the compound rest. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Large diameter
- 1
- Small diameter
- 0.5
- Length
- 4
- Taper per inch
- 0.125
- Taper angle deg
- 3.576334375
- Included angle deg
- 7.15266875
A tapered workpiece is a cone or frustum whose diameter changes steadily along the axis. Machinists, inspectors, and students describe that change as taper per inch and as a taper angle in degrees. This taper calculator accepts the large diameter, the small diameter, and the axial length of the tapered section, then returns both numbers from the same closed-form identities shops already use at the lathe. Taper per inch is the diameter difference divided by length. The taper angle is the half-angle from the centerline: the arctangent of taper per inch over two, converted to degrees.
How taper per inch and the taper angle are computed
On a right circular taper the diameter changes linearly from the large end to the small end over an axial length. When the three measures share inches, the result is the shop quantity people type into a taper per inch calculator; when they share millimetres the ratio is the same dimensionless slope and the angle is unchanged, because units cancel inside the quotient. The half-angle from the axis to the generating line is the arctangent of half the taper per inch, because the radius change is half the diameter change.
In degrees that identity is atan of (D minus d) over (two L), times one hundred eighty over pi. Equal diameters yield zero taper and a zero angle: a cylinder. A larger small end than large end yields a negative TPI, which is a reverse taper rather than a silent swap of the ends.
Where a taper per inch calculator earns its keep
Lathe hands and toolroom inspectors measure the two ends of a pin, a Morse shank, or a tapered bushing with a micrometer and need TPI plus the half-angle before they set a compound rest or stamp a certificate. Classroom and apprenticeship worksheets ask for taper per inch of a workpiece and for the corresponding angle in degrees; a deterministic oracle that rejects a non-positive length keeps auto-graders from treating a divide-by-zero as a clever negative slope.
Pin the rounded TPI and degrees in tests: one and one-half over four must stay 0.125 TPI. If you later need a Morse-number lookup, a taper-per-foot first-class field, or a millimetre-to-inch converter, treat those as separate search intents rather than overloading this taper page. The tool is a two-diameter plus length take-off, not a substitute for a taper-gauge set, a sine-bar inspection, or a CAD mass property on a part with a blended profile.
What you can do with it
Set a lathe compound from measured ends
Measure the large and small diameters and the axial length of a pin or bushing, then read taper per inch and the half-angle in degrees before setting the compound rest.
Inspect a tapered shank or bushing
Inspectors convert two micrometer readings and a length into TPI and included angle so a certificate matches the same identities the shop calculator shows.
FAQ
What happens if the length is zero or negative?
Length must be a positive number; taper per inch is undefined when you would divide by a non-positive length, and that failure is not billed.
Do the diameters have to be in inches?
No. Any shared linear unit works because the ratio is dimensionless and the angle does not depend on the unit. Inches are the usual shop language for “taper per inch”; millimetres of all three fields still yield the correct angle.
How do I calculate taper per inch?
Subtract the small diameter from the large one and divide by the length. Going from 1 to 0.5 over 4 gives 0.125 per inch; the same three measurements in millimetres give the same ratio and the same angle.
Is the angle the half-angle or the included angle?
Both are given. The taper angle is the half-angle from the centerline, the one you set on a compound rest; the included angle is twice that and matches drawings that quote the full opening of the cone.
What if the small end is larger than the large end?
You get a negative taper per inch, which means a reverse taper; the ends are not swapped for you. Equal diameters give zero taper and a zero angle, which is a plain cylinder.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/build/taper \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"large_diameter":1,"small_diameter":0.5,"length":4}'const res = await fetch("https://api.kit.forhosting.com/build/taper", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"large_diameter": 1,
"small_diameter": 0.5,
"length": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/build/taper",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"large_diameter": 1,
"small_diameter": 0.5,
"length": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/build/taper", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"large_diameter":1,"small_diameter":0.5,"length":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"large_diameter":1,"small_diameter":0.5,"length":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/build/taper", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"large_diameter": 1,
"small_diameter": 0.5,
"length": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "build.taper",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Limits
min_length | 1e-15 |
min_diameter | 0 |
max_abs | 1000000000000 |
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
A length that is not a positive number makes the ratio undefined, so the task fails with invalid_input instead of inventing a slope.
How taper per inch and the taper angle are computed
Taper per inch, often written TPI, is that diameter change per unit of length: large_diameter minus small_diameter, divided by length. The included cone angle is twice that half-angle and is reported as included_angle_deg so a drawing that quotes the full opening does not have to double the result by hand. Morse, Jacobs, Jarno, and Brown and Sharpe machine tapers are standardized TPI or taper-per-foot values; this endpoint does not look up a Morse number from a table, it evaluates the geometry you measured on the workpiece.
Input contract, units, and what happens when length is not positive
Missing fields fail the same way and name the field so a form can highlight length when someone types zero. All three measures must share one linear unit; mixing millimetres of diameter with inches of length would invent a slope that no drawing supports. Upper limits keep intermediate products finite: each field is at most 1e12 in magnitude, and length is at least 1e-15. Send three required numeric fields. large_diameter is a finite non-negative number for the diameter at the big end; aliases D, large, major_diameter, big_end, and major are accepted so a print that says major diameter does not need a rename. small_diameter is a finite non-negative number for the diameter at the little end; aliases d, small, minor_diameter, small_end, and minor are recognized, and a zero small end is a pointed full cone rather than an error. length is a finite positive number for the axial length of the tapered section; aliases L, taper_length, length_of_taper, and axial_length are accepted. Strings that parse as finite numbers are treated the same as JSON numbers. The hard rule this landing exists to enforce is that length must be a positive number: zero, a negative value, and a non-finite length fail with invalid_input because taper per inch would divide by a non-positive length and the angle would be undefined. On success the body echoes the cleaned diameters and length, publishes taper_per_inch as (D minus d) over L, taper_angle_deg as the half-angle in degrees, included_angle_deg as twice that half-angle, and formula as the string of those identities. Failed validation does not consume prepaid balance. Successful API calls use the published base of $0.002 per item.
Integration use cases
Grade taper-per-inch homework
Apprenticeship and LMS auto-graders check TPI and degrees from D, d, and L, and treat a non-positive length as invalid_input rather than a silent NaN.
Technical questions
How do I calculate taper per inch from two diameters and a length?
Send finite non-negative large_diameter and small_diameter plus a positive length in the same linear unit. Taper per inch is (large_diameter − small_diameter) / length.
How is the taper angle in degrees computed?
The half-angle from the axis is atan(taper_per_inch / 2) converted to degrees, which is atan((D − d) / (2L)) × 180/π. The included cone is twice that value and is returned as included_angle_deg.
What does the taper calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.