Oblique cone volume calculator
An oblique cone with base radius 3 and vertical height 5 holds about 47.1238898038 cubic units, exactly as much as an upright cone of the same size. Enter the radius and the perpendicular height to get the volume and base area. 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
- 5
- Volume
- 47.1238898038
- Base area
- 28.2743338823
An oblique circular cone has a circular base and an apex that is not necessarily centered above the base center, so the solid leans. Despite the lean, the volume is still one-third of the base area times the vertical height from the apex to the base plane. Cavalieri's principle makes this precise: solids with equal cross-sectional area at every corresponding height have equal volume, so an oblique cone matches a right circular cone that shares the same base radius and the same perpendicular height. Students meet this result when a textbook shows a slanted figure and asks for volume without measuring every generator. Architects, packaging designers, and CAD helpers need the same number when a drawing lists radius and story height rather than a full mesh.
What an oblique cone is and why Cavalieri fixes the volume
In Euclidean solid geometry a circular cone is the solid generated by joining every point of a disk to a fixed apex not in the plane of that disk. When the line from the apex to the center of the base is perpendicular to the base plane, the cone is right; when that axis is tilted, the cone is oblique.
The visual lean can trick the eye into thinking volume changed, but the measure that matters for volume is the perpendicular distance from the apex to the base plane, not the length of a slanted generator along the side surface. Cavalieri's principle, named after Bonaventura Cavalieri, states that solids with equal cross-sectional area at every corresponding height have equal volume.
Planes parallel to the base cut an oblique cone in disks whose radii scale linearly from the full base radius down to a point at the apex, just as in the right case, so the stack of infinitesimal slices has the same total volume. Algebraically this collapses to the elementary formula V equals one-third times pi times r squared times h, where r is the base radius and h is that orthogonal height.
Search intent clusters around phrases like oblique cone volume calculator, volume of a slanted cone, Cavalieri cone formula, and base radius times vertical height. People want a numeric volume they can check against a homework key or a bill of materials, not only a geometric argument that the solid is determined.
How this calculator validates input and applies V equals one-third pi r squared h
Both must be finite numbers that are not negative. When both measures pass, base area is pi times radius squared and volume is one-third of that product times height. Edge cases are explicit: missing fields, non-numeric strings, and negative measures all fail before any volume is returned.
Zero radius or zero height is accepted and yields volume zero, which is useful for boundary checks in scripts. You may use any consistent unit system; if radius is in centimeters and height in centimeters, volume is in cubic centimeters without conversion inside the engine. Do not pass a slanted generator as height unless you have already projected it to the orthogonal distance from the apex to the base plane, because Cavalieri requires the perpendicular measure, not the edge length of a leaning side.
Practical use, limits, pricing, and delivery channels
Use the browser tool when you want an immediate oblique-cone volume while checking solid-geometry homework, verifying a leaning hopper or pile volume from base radius and vertical height, or confirming a packaging sketch without installing a computer-algebra system. Results are fully deterministic: the same radius and height always yield the same volume. There is no language model, no external geometry service, and no randomness, so the capacity is safe to pin in continuous-integration checks and reproducible lesson materials.
Remember the contract limits: this tool does not compute lateral or total surface area, does not convert units, and does not accept a slant edge in place of vertical height. Surface formulas for an oblique cone depend on the lean angle and the generator lengths, which are outside the volume-only scope of this capability. Prefer explicit domain errors over opaque black-box widgets that cannot show why a negative measure is rejected when students paste a signed value from a partially filled form or a spreadsheet with inverted signs.
What you can do with it
Homework and exam oblique-cone drills
Students and graders confirm V=(1/3)πr²h for slanted cones from base radius and vertical height, including rejection when either dimension is negative.
FAQ
Which formula does the calculator use?
Cavalieri's principle: volume equals one-third of base area times perpendicular height, V = (1/3) π r² h. An oblique cone has the same volume as a right circular cone with the same base radius and the same vertical height.
What happens if radius or height is negative?
Both radius and height must be finite and not negative; negative dimensions are rejected and those calls are not charged.
Is the oblique cone volume calculator free?
The browser tool runs free on your device.
Can I use a slant edge as the height?
No. Height must be the vertical (perpendicular) distance from the apex to the base plane. If you only know a generator length, project it to the orthogonal height first; Cavalieri requires that perpendicular measure.
Why does leaning the cone not change its volume?
By Cavalieri's principle, solids with equal cross-sections at every height have equal volume. Slices parallel to the base of a leaning cone are the same disks as in an upright cone with the same radius and height, so one third of π r² h still applies.
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/oblique-cone \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"radius":3,"height":5}'const res = await fetch("https://api.kit.forhosting.com/geom/oblique-cone", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"radius": 3,
"height": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/oblique-cone",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"radius": 3,
"height": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/oblique-cone", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"radius":3,"height":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"radius":3,"height":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/oblique-cone", 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": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.oblique_cone",
"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
This capacity accepts non-negative radius and height, rejects negative dimensions, and returns volume with the Cavalieri method tag as structured JSON.
What an oblique cone is and why Cavalieri fixes the volume
Delivering volume, radius, height, base area, method, and formula as JSON makes the result easy to paste into notes, diff in automated tests, or feed into a larger pipeline without reimplementing the one-third product in every language you teach or deploy across lesson platforms and packaging tools worldwide.
How this calculator validates input and applies V equals one-third pi r squared h
Send a JSON body with two required fields: radius and height. The engine parses each field as a number or a numeric string, rejects empty or non-finite values, and then enforces non-negativity: a negative radius or a negative height fails with an invalid_input message that names the bad field and states that dimensions cannot be negative. Results are rounded to a stable decimal precision so the same input always yields the same JSON across JavaScript engines. The response echoes the validated radius and height, reports base_area, tags method as cavalieri, and records the formula string V equals one-third pi r squared h for transparency in lesson materials.
Practical use, limits, pricing, and delivery channels
Each successful request is one billable item at $0.002; failed validation does not consume balance. Accepted inputs must not be negative; negative radius and negative height are rejected with a clear invalid_input message. Wire the endpoint into solid-geometry drills, contest answer checkers, CAD preprocessing steps that already know base radius and orthogonal height, or floating worksheets that list only those two measures.
Integration use cases
Deterministic fixtures for solid-geometry tests
Pipelines that seed unit tests or interactive cone demos can call a pure Cavalieri oracle instead of reimplementing the one-third product in every language.
Architecture and packaging from radius and story height
Notes that list only base radius and orthogonal height recover volume as structured JSON for reports, size charts, or lightweight CAD helpers without full meshes.
Technical questions
How do I compute the volume of an oblique cone with this API?
Send a JSON body with non-negative radius and height to geom.oblique_cone. The response returns volume ((1/3)πr²h), base_area, the echoed inputs, method cavalieri, and the formula V = (1/3) π r² h.