Paraboloid volume calculator
A paraboloid with base radius 3 and height 4 holds 56.5486677646 cubic units, exactly half of the 113.0973355292 in the cylinder around it. Enter radius and height to size dishes, tank bottoms or nose cones. 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
- Base area
- 28.2743338823
- Cylinder volume
- 113.0973355292
- Volume
- 56.5486677646
A circular paraboloid of revolution is the solid obtained by rotating a parabolic arc about its axis so that every cross-section parallel to the base is a disk whose radius grows with the square root of axial distance from the vertex. Once the base radius at the open end and the height from vertex to base plane are known and strictly positive, the classical volume formula is fully determined: half pi times the square of the radius times the height, which is exactly half the volume of the right circular cylinder that shares the same base and height. Students meet this half-cylinder identity in solid-geometry units on solids of revolution; engineers and makers need the same number for parabolic dishes, nose-cone approximations, tank bottoms, and packaging notes that list only r and h.
What a paraboloid of revolution is and why the volume is half the cylinder
In Euclidean solid geometry a circular paraboloid of revolution is generated by rotating a parabola about its axis of symmetry. At the vertex the cross-section collapses to a point; at axial distance z from the vertex the radius of the circular slice scales with the square root of z, reaching the full base radius r when z equals the height h. The solid is therefore completely determined, up to congruence, by the two positive lengths r and h.
Integrating disk areas along the axis yields the closed form volume equal to one-half pi r squared h. The same result is often stated as a comparison with the enclosing right circular cylinder of radius r and height h, whose volume is pi r squared h: the paraboloid occupies exactly half of that cylinder.
This half-cylinder identity is the memorable homework and exam fact that people search for under phrases such as paraboloid volume calculator, volume of a paraboloid of revolution, half cylinder paraboloid, and volume from base radius and height.
How this calculator validates input and applies the closed-form formula
Missing fields, empty strings, non-numeric tokens, zero, and negative measures all fail before any volume is returned. When both measures pass positivity checks, the engine computes the circular base area as pi times radius squared, the enclosing cylinder volume as base area times height, and the paraboloid volume as one half of that cylinder volume, which is the same as one-half pi r squared h.
Edge cases are explicit: a negative radius or height is rejected with a clear message that the measure must be positive; a zero height is refused because a solid of vanishing height is not a valid paraboloid under this contract; non-finite intermediate results after extreme magnitudes are also refused. You may use any consistent unit system for radius and height; area and volume are expressed in the corresponding square and cubic units without unit conversion inside the engine.
FAQ
Which formula does the calculator use?
Volume is (1/2)·π·r²·h. That is exactly half the enclosing cylinder volume π·r²·h of the same base radius and height.
What happens if the radius or height is not positive?
A paraboloid of revolution requires strictly positive base radius and height; non-positive values are rejected and those calls are not charged.
Is the paraboloid volume calculator free?
The browser tool runs free on your device.
Does this handle elliptic paraboloids with two different semi-axes?
No. This capacity solves the circular solid of revolution only: one base radius and one height. Elliptic bases with unequal semi-axes are out of scope.
How does the radius of a paraboloid change along its height?
It grows with the square root of the distance from the vertex. At the vertex the cross-section is a single point, and it reaches the full base radius at the open end.
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/paraboloid \
-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/paraboloid", {
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/paraboloid",
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/paraboloid", 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/paraboloid", 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.paraboloid",
"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 positive radius and height, rejects non-positive measures, and returns volume together with base area and the enclosing cylinder volume as structured JSON so you can check the half-cylinder relation at a glance.
What a paraboloid of revolution is and why the volume is half the cylinder
Delivering volume, cylinder_volume, and base_area together lets a student or engineer verify the relation without re-deriving the integral, and structured JSON makes the result easy to paste into notes, answer keys, CAD helpers, or automated tests without reimplementing pi multiplications in every language.
How this calculator validates input and applies the closed-form formula
Send a JSON body with both radius and height. Each field must be a positive finite number; the engine accepts ordinary JSON numbers or numeric strings that parse to finite values. Results are rounded to a stable decimal precision so the same input always yields the same JSON across JavaScript engines.
Practical use, limits, pricing, and delivery channels
Use the browser tool when you want an immediate paraboloid volume while checking homework, sizing a parabolic tank bottom or dish approximation, verifying a packaging fill estimate, or confirming a workshop drawing without installing a computer-algebra system. Results are fully deterministic: the same pair always yields the same base area, cylinder volume, and paraboloid 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. Each successful request is one billable item at $0.002; failed validation does not consume balance. Accepted inputs must be strictly positive for both radius and height; non-positive values always fail with invalid_input. Wire the endpoint into solid-geometry drills, contest answer checkers, tank-capacity worksheets that list only base radius and height, or floating preprocessing steps where you already prefer explicit domain errors over opaque black-box widgets that cannot show why a non-positive measure is rejected.
Integration use cases
Homework and exam paraboloid volume drills
Students and graders confirm V = (1/2)πr²h from base radius and height, including rejection when radius or height is not positive, and verify the half-cylinder identity against the returned cylinder_volume.
Deterministic fixtures for solids-of-revolution tests
Pipelines that seed unit tests or interactive revolution demos can call a pure paraboloid oracle instead of reimplementing (1/2)πr²h in every language or framework.
Dishes, tank bottoms, and packaging from r and h
Shop or logistics notes that list only a base radius and axial height recover paraboloid volume and the enclosing cylinder volume as structured JSON for capacity charts or lightweight CAD helpers.
Technical questions
How do I compute paraboloid volume with this API?
Send a JSON body with positive radius and height to geom.paraboloid. The response returns radius, height, base_area, cylinder_volume, and volume.