Conical frustum slant height calculator
Get the slant height of a truncated cone from its two radii and height, so radii 3 and 4 with a height of 5 give √26, about 5.099. Only the side length, with no volume or area to wade through. Free in your browser.
- Free
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- No sign-up
- Your data stays on your device
Run — free
Worked example
- Radius1
- 3
- Radius2
- 4
- Height
- 5
- Slant height
- 5.0990195136
- Radius diff
- -1
The slant height of a conical frustum is the length of the straight generator that runs along the lateral surface between the two parallel circular bases. Students meet this measure when textbooks ask for the apothem of a truncated cone, when they unrolled the side into a sector of an annulus, or when they need the hypotenuse of the right triangle formed by the perpendicular height and the difference of the base radii. Shop drawings for hoppers, funnels, lamp shades, and tapered tanks often list top radius, bottom radius, and vertical rise and expect the generator length next. This conical frustum slant height calculator accepts radius1, radius2, and height, rejects any negative dimension, and returns the closed-form slant height L equals the square root of height squared plus the squared difference of the radii.
What frustum slant height means and why the hypotenuse formula works
A right circular conical frustum is the solid left when a plane parallel to the base cuts a right circular cone short of the apex, leaving two parallel circular faces and a tapered lateral surface. The perpendicular height is the axial distance between those faces; it is not the same as the edge that runs on the side. If you slice the frustum with a plane that contains the axis, you obtain a trapezoid whose non-parallel sides are equal generators.
Drop a perpendicular from the end of the shorter base to the longer base and you isolate a right triangle whose vertical leg is the height h, whose horizontal leg is the absolute difference of the two radii, and whose hypotenuse is exactly the generator length L. Pythagoras therefore gives L equals the square root of h squared plus (R minus r) squared, without needing the cone’s original apex or the full height of the parent solid.
Search intent clusters around phrases such as conical frustum slant height calculator, truncated cone generator length, frustum apothem, and slant height of a frustum from two radii. People want a single numeric length they can check against an answer key, a material cut list, or a tank fabrication note, not a multi-metric dump of volume and surface area when only the side length is required.
What you enter and what the result shows
Every dimension must be non-negative: a negative radius1, radius2, or height is rejected immediately with a clear message that frustum dimensions cannot be negative, and those validation failures are not billed. Zero is allowed on any field so that a full cone (one radius zero), a cylinder side equal to the height (equal radii), or a flat annular limit (height zero with unequal radii) remain expressible under the same contract.
Example: radius1 equals three, radius2 equals four, and height equals five yields slant height approximately five point zero nine nine, because the radius difference is negative one and the hypotenuse of a three-four-five style right triangle with legs five and one is the square root of twenty-six. Units are whatever consistent length system you already use; the engine never converts meters to inches.
What you can do with it
Homework checks for frustum generator length
Students and tutors verify L = √(h²+(R−r)²) from two radii and height, including rejection when any dimension is negative.
FAQ
Which formula does the calculator use?
Slant height (generator length) is L = √(h² + (R − r)²), the hypotenuse of the right triangle formed by the perpendicular height and the difference of the two base radii.
What happens if a dimension is negative?
Frustum radii and height must not be negative; those validation failures are not charged.
Is the conical frustum slant height calculator free?
The browser tool runs free on your device.
Does this return volume or surface area as well?
No. This capacity returns only slant height and related echo fields.
Is the slant height the same as the perpendicular height?
No. The height is measured straight along the axis between the two bases, while the slant height runs along the side. It is the hypotenuse of a right triangle whose legs are the height and the difference of the radii.
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/geom/frustum-slant-height \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"radius1":3,"radius2":4,"height":5}'const res = await fetch("https://api.kit.forhosting.com/geom/frustum-slant-height", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"radius1": 3,
"radius2": 4,
"height": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/frustum-slant-height",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"radius1": 3,
"radius2": 4,
"height": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/frustum-slant-height", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"radius1":3,"radius2":4,"height":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"radius1":3,"radius2":4,"height":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/frustum-slant-height", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"radius1": 3,
"radius2": 4,
"height": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.frustum_slant_height",
"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.
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
What frustum slant height means and why the hypotenuse formula works
Returning slant_height with the echoed radii, height, signed radius difference, and formula string makes the result easy to paste into notes, pin in automated tests, or feed into a larger pipeline without reimplementing square roots in every language you teach or deploy.
Input contract, validation rules, and response fields
Send a JSON body with three required fields: radius1, radius2, and height. Each value may be a JSON number or a numeric string; missing fields, empty strings, and non-finite values fail with invalid_input naming the field. On success the body echoes the three inputs rounded for stable comparison, returns slant_height from the closed form L equals square root of h squared plus (R minus r) squared, includes radius_diff as the signed difference R minus r for clients that want the horizontal leg without recomputing it, and attaches a short formula string plus method frustum_slant_height for display and logging.
Where a dedicated slant-height endpoint earns its keep
Classrooms and tutoring platforms use a conical frustum slant height calculator so learners can check the generator length without also computing volume when the homework item only asks for the side. Fabrication helpers for hoppers, funnels, and tapered sheet-metal parts often know top radius, bottom radius, and rise from a drawing and need only the cut length along the generator before unrolling the lateral surface. Content sites that rank for long-tail queries such as conical frustum slant height calculator benefit from a focused landing whose free browser path matches the paid automation path byte-for-byte on the same pure module. Exam-prep generators need a deterministic oracle that marks negative dimensions as invalid_input rather than silently returning NaN or a complex value that a buggy grader might accept. Pair the response with your own unrolling math if you need sector arc length or pattern angle; this endpoint solves for slant height only and does not emit drawings or area totals.
Integration use cases
Deterministic fixtures for solid-geometry tests
Pipelines that seed unit tests or interactive frustum demos call a pure slant-height oracle instead of reimplementing Pythagoras in every language.
Shop generator length from measured top and bottom radii
Fabrication notes that list top radius, bottom radius, and rise recover the cut length along the side as structured JSON for cut lists or lightweight CAD helpers.
Technical questions
How do I calculate conical frustum slant height with this API?
Send a JSON body with radius1, radius2, and height to geom.frustum_slant_height. The response returns slant_height as √(h²+(R−r)²) plus echoed inputs and the formula string.