Slant height calculator
Get a cone's slant height from its base radius and vertical height, so radius 3 and height 4 give exactly 5, the side length you cut for a funnel, a party hat or a conical roof. Free in your browser.
- Free
- Runs in your browser
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- 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. That length is the hypotenuse of the right triangle formed by the base radius and the perpendicular height, so the Pythagorean theorem supplies a closed form: l equals the square root of r squared plus h squared. Students meet this quantity when a textbook asks for the generator of a cone, when they unroll the side into a circular sector, or when a shop drawing lists only radius and rise and they need the cut length of a funnel, party hat, or conical roof. This slant height calculator accepts a positive base radius and a positive vertical height, rejects any non-positive dimension, and returns the deterministic generator length.
What slant height means and why Pythagoras is the whole algorithm
A right circular cone has a circular base and an apex that sits on the axis meeting the base center at a right angle. The vertical height is that axial distance from the base center to the 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. The Pythagorean theorem therefore gives l equals the square root of r squared plus h squared, without needing the lateral area, the volume, or any unrolling angle. People want that one hypotenuse, not a multi-metric dump of volume and surface area when only the side length is required.
The dedicated landing targets the query slant height, distinct from pages that bury the generator among volume and area fields. Because the axial triangle is always right-angled for a right circular cone, there is no iterative solver, no angle conversion, and no special function beyond a single square root of a sum of squares.
What to enter and what you get back
Every dimension must be strictly positive: a zero or negative radius or height is rejected immediately with a clear message that radius and height must be positive, and those validation failures are not billed. Zero is not treated as a degenerate disk or needle on this landing, because a proper right circular cone has a positive base and a positive rise. 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 or feet to centimetres. Keep both numbers finite and well below the internal magnitude ceiling so the squared sum stays inside ordinary floating-point range.
Where the slant height comes in handy
Classrooms and tutoring platforms use a 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 the query slant height benefit from a focused landing whose free browser path matches the paid automation path byte-for-byte on the same pure module. The capability is a Pythagorean landing: when you already have r and h and you only need the hypotenuse of the axial triangle, this is the precise contract.
What you can do with it
Homework checks for cone generator length
Students and tutors verify l = √(r²+h²) from a positive base radius and vertical height, including rejection when either dimension is not positive.
Sheet-metal and roofing cut lists
Fabricators convert a drawing that lists radius and rise into the generator length needed to unroll a conical hopper, funnel, or roof panel.
FAQ
What happens if radius or height is zero or negative?
A proper cone on this landing needs a positive base radius and a positive vertical height; those failures are not billed.
Does the calculator convert units?
No. Radius, height, and slant height share whatever consistent length unit you already use. Convert metres to inches yourself before calling.
Is the slant height the same as the perpendicular height?
No. Height is the axial rise from the base center to the apex. Slant height is the generator along the side, always strictly longer than height when the radius is positive.
How do I find the slant height of a cone with radius 3 and height 4?
Square the radius and the height, add them and take the square root of the sum. Radius 3 and height 4 give a slant height of exactly 5, the classic right triangle hidden inside the cone.
Does it also give the volume or surface area of the cone?
No. It returns only the slant height, the straight line along the side from the rim of the base to the tip. For area or volume, plug that length into the usual cone formulas yourself.
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/math/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/math/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/math/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/math/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/math/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": "math.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 slant height means and why Pythagoras is the whole algorithm
Returning slant_height with the echoed radius, height, 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 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 the square root of r squared plus h squared, and attaches a short formula string 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, area totals, or volume.
Where a slant-height endpoint earns its keep
Exam-prep generators need a deterministic oracle that marks non-positive dimensions as invalid_input rather than silently returning NaN or a complex value that a buggy grader might accept. When you later need volume or surface as well, combine the returned slant height with the usual closed forms in your own notebook rather than asking this endpoint to invent extra fields.
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.
Technical questions
How do I compute the slant height of a right circular cone?
Send positive finite numbers radius and height. The API returns l = √(r²+h²), the hypotenuse of the axial right triangle, plus the echoed inputs and the formula string.
What does the slant height calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.