Cone slant height calculator
Find the slant height of a cone, the straight edge from the base rim up to the tip, from its radius and height; radius 3 and height 4 give exactly 5. Handy before cutting a funnel, a party hat or a conical roof. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Radius
- 3
- Height
- 4
- Slant height
- 5
The slant height of a right circular cone is the straight-line distance from any point on the circular base rim up the lateral surface to the apex. Students meet this length when textbooks ask for the generator of a cone, when they unroll the side into a sector of a circle, or when they need the hypotenuse of the right triangle formed by the perpendicular height and the base radius. Shop drawings for funnels, party hats, tapered hoppers, and conical roofs often list radius and vertical rise and expect the generator next. This cone slant height calculator accepts radius and height, rejects any negative dimension, and returns the closed-form slant height l equals the square root of radius squared plus height squared.
What cone slant height means and why the hypotenuse formula works
A right circular cone has a circular base and an apex lying on the axis that meets the base center at a right angle. The perpendicular height is that axial distance from base center to apex; it is not the same as the edge that runs on the side. If you slice the cone with a plane that contains the axis, you obtain an isosceles triangle whose equal legs are the generators.
Half of that axial section is a right triangle whose vertical leg is the height h, whose horizontal leg is the base radius r, and whose hypotenuse is exactly the generator length l. Pythagoras therefore gives l equals the square root of r squared plus h squared, without needing the lateral area, the volume, or any unrolling angle. Search intent clusters around phrases such as cone slant height calculator, cone generator length, cone apothem, and slant height of a cone from radius and height.
People want a single numeric length they can check against an answer key, a material cut list, or a fabrication note, not a multi-metric dump of volume and surface area when only the side length is required. The dedicated landing targets the long-tail query cone-slant-height-calculator, distinct from multi-metric cone volume pages that bury the generator among volume and area fields.
What you enter and what comes back
Every dimension must be non-negative: a negative radius or height is rejected immediately with a clear message that cone dimensions cannot be negative, and those validation failures are not billed. Zero is allowed on either field so that a flat disk (height zero with positive radius, slant equal to the radius) or a degenerate needle (radius zero with positive height, slant equal to the height) remain expressible under the same contract. Example: radius equals three and height equals four yields slant height exactly five, the classic three-four-five right triangle.
Another common check is radius five and height twelve, which yields thirteen. 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 cone generator length
Students and tutors verify l = √(r²+h²) from base radius and height, including rejection when any dimension is negative.
FAQ
Which formula does the calculator use?
Slant height (generator length) is l = √(r² + h²), the hypotenuse of the right triangle formed by the perpendicular height and the base radius.
What happens if a dimension is negative?
Cone radius and height must not be negative; those validation failures are not charged.
Is the cone 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. For volume and surface metrics of a cone, use a multi-metric cone calculator such as geom.cone.
Can the radius or the height be zero?
Yes. A height of zero describes a flat disk, so the slant height equals the radius; a radius of zero describes a needle, so it equals the height. Only negative values are rejected.
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/cone-slant-height \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"radius":3,"height":4}'const res = await fetch("https://api.kit.forhosting.com/geom/cone-slant-height", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"radius": 3,
"height": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/cone-slant-height",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"radius": 3,
"height": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/cone-slant-height", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"radius":3,"height":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"radius":3,"height":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/cone-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
{
"radius": 3,
"height": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.cone_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 cone slant height means and why the hypotenuse formula works
Returning slant_height with the echoed radius, height, formula string, and method name 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 two required fields: radius 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 two inputs rounded for stable comparison, returns slant_height from the closed form l equals square root of r squared plus h squared, and attaches a short formula string plus method cone_slant_height for display and logging. 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.
Where a dedicated cone slant-height endpoint earns its keep
Classrooms and tutoring platforms use a cone 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 funnels, conical roofs, and tapered sheet-metal parts often know base radius and rise from a drawing and need only the cut length along the generator before unrolling the lateral surface into a sector. Content sites that rank for long-tail queries such as cone 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. The capacity complements multi-metric solvers such as geom.cone: when you need volume and surface as well, use those tools; when the search intent and the workflow only need the hypotenuse of the axial triangle, this endpoint is the precise contract.
Integration use cases
Deterministic fixtures for solid-geometry tests
Pipelines that seed unit tests or interactive cone demos call a pure slant-height oracle instead of reimplementing Pythagoras in every language.
Shop generator length from measured radius and rise
Fabrication notes that list base radius and vertical rise recover the cut length along the side as structured JSON for cut lists or lightweight CAD helpers.
Technical questions
How do I calculate cone slant height with this API?
Send a JSON body with radius and height to geom.cone_slant_height. The response returns slant_height as √(r²+h²) plus echoed inputs and the formula string.