Oblique cylinder calculator
A cylinder with radius 3 and perpendicular height 10 holds about 282.7433 cubic units, however far it leans. Enter the radius, the height and, if you have it, the slanted edge to get the volume plus the lateral, base and total areas. Free in your browser.
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Worked example
- Base radius (r)
- 3
- Perpendicular height (h)
- 10
- Generator length
- 12
- Volume
- 282.7433
- Lateral surface area
- 113.0973
- Base area
- 28.2743
- Total surface area
- 169.646
Compute the volume and surface areas of an oblique circular cylinder from its radius r, perpendicular height h and optional slant generatrix. Returns volume = π·r²·h, lateral area, base area and total area.
Why a leaning cylinder keeps the same volume
An oblique cylinder is a circular cylinder whose axis is tilted, so the top disk sits offset from the bottom one instead of straight above it. Its volume is still π·r²·h, where r is the base radius and h is the perpendicular height between the two parallel base planes, not the length of the slanted axis. Cavalieri's principle explains why: slide every thin horizontal slice sideways and none of them changes its area, so the stack holds exactly as much as a right cylinder with the same base and height. With a radius of 3 and a height of 10 the volume is about 282.7433 cubic units.
Surface areas, uses and the usual mistakes
For the surface figures the calculator also uses the slant length, the generatrix, which is the tilted side edge. Leave it out and it is taken as equal to the height, the upright case; give it and it must be at least as long as the height, because a slanted edge can never be shorter than the vertical gap between the bases. Engineers use this to size tilted tanks, pipes and columns whose fill height is known, and teachers use it to demonstrate Cavalieri's principle. The classic slips are entering the axis length as the height and typing the diameter where the radius goes.
Volume by Cavalieri's principle
An oblique cylinder is a circular cylinder whose axis is tilted, so its top and bottom disks are offset rather than stacked vertically. Its volume is still volume = π·r²·h, where r is the base radius and h is the perpendicular height between the two parallel base planes, not the slanted axis length. This follows from Cavalieri's principle: sliding the horizontal cross-sections sideways does not change their areas, so an oblique cylinder holds exactly as much as a right cylinder with the same base and the same perpendicular height.
The calculator takes r and h as required numeric fields, both strictly positive, and reports volume alongside the surface figures.
Lateral, base and total areas
The surface outputs use the slant length through the generatrix field, the length of the tilted lateral edge. If you omit generatrix it defaults to the perpendicular height h, which recovers the right-cylinder case where the edge stands vertical. The returned pi field carries Math.PI at full double precision and is never rounded, so you can reproduce the intermediate steps.
What you can do with it
Engineering and volume estimates
Size tilted tanks, pipes or columns where the axis is not vertical but the fill height is known.
Geometry teaching
Demonstrate Cavalieri's principle: the same π·r²·h volume whether the cylinder is right or oblique.
FAQ
Does the tilt change the volume?
No. By Cavalieri's principle the volume stays π·r²·h using the perpendicular height h, regardless of the generatrix.
How is rounding controlled?
The decimals field, an integer 0 to 8 (default 4), rounds each area and the volume half-up once from the raw value.
Which height do I enter for a tilted cylinder?
The perpendicular height, meaning the straight vertical distance between the two parallel bases. The tilted axis is longer, and using its length would make the volume too large.
What is the generatrix, and what happens if I leave it out?
It is the length of the slanted side edge, used for the surface areas. If you leave it out, it is set equal to the perpendicular height, which is the case of an ordinary upright cylinder.
Can the slanted edge be shorter than the height?
No. A tilted edge always spans at least the vertical gap between the bases, so a value shorter than the height is refused instead of producing impossible surface areas.
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/oblique-cylinder \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"r":3,"h":10}'const res = await fetch("https://api.kit.forhosting.com/math/oblique-cylinder", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"r": 3,
"h": 10
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/oblique-cylinder",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"r": 3,
"h": 10
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/oblique-cylinder", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"r":3,"h":10}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"r":3,"h":10}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/oblique-cylinder", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"r": 3,
"h": 10
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.oblique_cylinder",
"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
Volume by Cavalieri's principle
For the worked example r = 3 and h = 10 the volume is π·9·10 ≈ 282.7433, independent of how far the top disk is shifted horizontally by the tilt.
Lateral, base and total areas
The tool computes lateral_area = π·r·generatrix, base_area = π·r² for one circular end, and total_area = lateral_area + 2·base_area, adding both caps to the side. With r = 3, h = 10 and generatrix = 12 the tool returns lateral_area ≈ 113.0973, base_area ≈ 28.2743 and total_area ≈ 169.646, each derived from the raw product before rounding rather than from another rounded field.
Inputs, rounding and validation
The generatrix, when supplied, must be a finite number that is at least h, because a slanted lateral edge can never be shorter than the vertical distance between the bases; a shorter value returns "generatrix inválido". Non-finite intermediate results are rejected before output. Both r and h must be finite numbers greater than zero, or the tool raises a typed invalid_input error naming the field, for example "r inválido" or "h inválido". The optional decimals field is an integer from 0 to 8, defaulting to 4, and each of volume, lateral_area, base_area and total_area is rounded half-up exactly once from its raw intermediate value.
Integration use cases
Batch surface calculations via API
Send many r, h and generatrix triples as JSON and get volume with lateral, base and total areas for each.
Technical questions
What is the generatrix?
The slant lateral edge length, used for lateral_area = π·r·generatrix. It defaults to h and must be at least h.
What does it output?
volume, lateral_area, base_area, total_area (lateral + two bases), plus pi at full precision, with your r, h and generatrix.
Is it free?
Yes in the browser as a tier A widget. The deterministic API costs $0.002 per successful call; failed tasks are not charged.