Bowl segment calculator
A bowl of radius 10 cm filled 4 cm deep holds 416π/3 cm³, with a liquid surface of rim radius 8 cm, shown in litres and US gallons as well. Use it to scale recipes or size serving bowls. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Radius
- 10 cm
- Height
- 4
- Diameter
- 20 cm
- Rim radius
- 8
- Volume
- 435.6341812978
- Volume m3
- 0.0004356342
Given those two lengths and the unit you measured them in, this calculator returns the enclosed volume, the curved interior surface that is wetted, the circular free surface of the liquid, and the same capacity expressed in litres and US gallons. No network, no sampling, and no clock.
How bowl-segment volume, surface, and capacity are computed
Treat the bowl as a portion of a sphere of radius R and measure the liquid from the lowest interior point up to the free surface; that perpendicular distance is the fill height h. The solid of liquid is then a spherical cap. Its volume is one-third pi times h squared times three R minus h: volume equals (1/3) π h² (3R − h), in cubic units of the length you sent.
When h equals R you recover a full hemispherical bowl: volume (2/3) π R³ and curved area 2 π R². When h equals the diameter 2R you recover a completely filled sphere. A classic check is radius ten centimetres and fill height four centimetres, which yields volume 416π/3 cubic centimetres, a rim radius of eight centimetres, and the matching litre and gallon figures.
Because the path uses only multiplication, a fixed fraction one-third, square root, and Math.PI, it is deterministic, O(1), and free of iterative solvers.
What you send and what you get back
Send three required fields. radius is the sphere radius of the bowl in your chosen length unit; it must be a finite number strictly greater than zero. height is the fill height measured from the bottom, in the same unit, strictly greater than zero and not greater than the bowl diameter (twice the radius). unit is the length unit for both dimensions and must be one of cm, mm, in, m, or ft; the comparison is case-insensitive and common spellings such as inches, centimetres, metres, and feet map onto those five keys.
Unrecognized units fail rather than silently defaulting, because a guessed unit would hide a gallon error of two or three orders of magnitude. Example: radius ten, height four, unit cm is the textbook shallow-bowl check. Radius five and height twelve fails because twelve exceeds the diameter ten.
Wire the free browser tool for a quick punch-bowl check, then call the identical contract from scripts for bulk SKUs without reimplementing the cap formulas.
Where a bowl segment calculator earns its keep
Commercial kitchens and catering desks reach for a bowl segment calculator when a mixing bowl, punch bowl, or serving vessel is spherical or close enough that a cap formula beats a cylinder guess. Knowing the inner radius and how high you will fill it yields litres for a recipe scale-up and US gallons for a beverage station that is specified in gallons.
Lab glassware includes hemispherical evaporating dishes and round-bottom flasks used as open bowls; technicians who must not overfill past a sagitta can fail closed when height would exceed the diameter instead of trusting a spreadsheet that silently applies a hemisphere formula at the wrong depth. Garden and landscape work uses spherical bird baths, fountain bowls, and irrigation basins; volume in litres sizes a pump or a treatment dose, while curved area estimates liner or coating.
If you only need cubic length units with no litre or gallon conversion, that is a different search intent (a spherical-cap geometry landing) rather than this bowl-capacity page. Keep requests small, pin the rounded litres and gallons in tests, and you get reproducible capacity for recipes, quotes, and edge workers without standing up a CAD kernel of your own.
What you can do with it
Kitchen mixing and serving bowls
Convert a spherical mixing bowl’s inner radius and fill height into litres and US gallons so a recipe scale-up matches the vessel you actually own.
Catering punch bowls and drink stations
Size a punch or sangria bowl from radius and how high you will fill it, then quote beverage volume in gallons without a cylinder approximation.
Lab dishes and fountain basins
Fill a hemispherical evaporating dish, bird bath, or fountain bowl to a measured sagitta and get capacity plus wetted area for dose or coating estimates.
FAQ
How do I compute bowl-segment volume from radius and fill height?
Send positive radius and height in the same length unit, plus unit as cm, mm, in, m, or ft. Volume is (1/3)πh²(3R−h); the response also returns litres and US gallons.
Are the gallons US or imperial?
US liquid gallons. One US gallon is exactly 3.785411784 litres (231 cubic inches). Imperial gallons are not returned.
Which length units can I send?
cm, mm, in, m, or ft, as the unit field. Radius and height must share that unit. Capacity is always litres and US gallons regardless of the input unit.
What happens if the fill height is more than the bowl diameter?
The calculation is refused, because liquid cannot stand higher than a full sphere. The fill height must be greater than zero and at most twice the radius; a height equal to the diameter means a completely filled sphere.
What volume does a bowl filled to its radius hold?
Filling to a height equal to the radius gives a full hemisphere: two-thirds of π times the radius cubed, with a curved wetted area of 2πR². It is a handy check for any hemispherical bowl.
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/build/bowl-segment \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"radius":10,"height":4,"unit":"cm"}'const res = await fetch("https://api.kit.forhosting.com/build/bowl-segment", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"radius": 10,
"height": 4,
"unit": "cm"
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/build/bowl-segment",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"radius": 10,
"height": 4,
"unit": "cm"
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/build/bowl-segment", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"radius":10,"height":4,"unit":"cm"}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"radius":10,"height":4,"unit":"cm"}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/build/bowl-segment", 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": 10,
"height": 4,
"unit": "cm"
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "build.bowl_segment",
"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
A spherical bowl segment is the liquid that sits in a spherical bowl when you fill it to a measured height: a spherical cap of height h cut from a sphere of radius R. Fill height greater than the bowl diameter is rejected with invalid_input instead of a nonsense volume.
How bowl-segment volume, surface, and capacity are computed
The curved interior that is wetted—the spherical zone, not the circular top—is two pi times R times h: curved_surface_area equals 2 π R h. The free surface is a circle whose radius is the square root of h times (2R minus h); free_surface_area is pi times that radius squared, and total_surface_area is the sum of the curved and free parts. Lengths convert to metres with exact factors (a centimetre is 0.01 m, an inch is 0.0254 m) so capacity is independent of the input unit: volume_liters is cubic metres times one thousand, and volume_gallons is that litre figure divided by 3.785411784, the exact litre content of one US liquid gallon. Huge magnitudes can still overflow IEEE-754; when intermediates are non-finite the request fails with a clear overflow message instead of returning Infinity in JSON.
What you send and what you get back
Missing fields, empty strings, non-numeric text, NaN, Infinity, non-positive radius or height, and height exceeding the diameter are rejected with invalid_input and are not charged. On success the JSON echoes cleaned radius, height, diameter (2×radius), and the canonical unit; rim_radius of the circular free surface; volume in cubic input units; volume_m3; volume_liters; volume_gallons as US liquid gallons; curved_surface_area, free_surface_area, and total_surface_area in square input units; plus formula and method so logs can document the algorithm. API pricing is the published base of $0.002 per successful item.
Where a bowl segment calculator earns its keep
Curriculum authors for culinary math and introductory solid geometry want one JSON shape in the browser and the API so worksheets and auto-graders stay aligned. Because this capability is a pure parse engine, API pricing is the published base of $0.002 per successful item and failed validation does not consume balance.
Technical questions
What happens if fill height exceeds the bowl diameter?
The task fails with invalid_input. A spherical bowl segment taller than the diameter (2×radius) is not defined, so the request is not billed.
What does the bowl segment calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.