Parabolic segment area calculator
A parabolic segment with base 6 and height 3 has an area of 12: two thirds of base times height, and four thirds of the triangle of area 9 inside it. Enter both lengths for arches, reflectors or tank ends. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Base
- 6
- Height
- 3
- Area
- 12
- Triangle area
- 9
Given only the length of that base chord and the perpendicular height from the chord’s midpoint out to the curve, Archimedes proved that the enclosed area is exactly two-thirds of the product of base and height. Students meet this result in calculus prefaces and history-of-math units; surveyors, boat designers, and makers need the same closed form when a parabolic arch, reflector profile, or tank end is cut by a flat plate.
What a parabolic segment is and why Archimedes’ formula fixes its area
In plane geometry a parabola is the locus of points equidistant from a focus and a directrix. Cut that curve with a straight chord and the finite region between chord and arc is the parabolic segment. Unlike a circular segment, which needs a radius and a central angle, a parabolic segment of a given chord is fully determined by the base length of the chord and the sagitta—the perpendicular distance from the midpoint of the chord to the parabola.
Archimedes’ quadrature of the parabola shows that this area equals four-thirds the area of the inscribed triangle that shares the same base and height, which simplifies at once to A equals two-thirds times base times height. At zero height the chord sits on a degenerate flat arc and the area is zero; at zero base the figure collapses to a line. Search intent clusters around phrases such as parabolic segment area, area of a parabolic segment calculator, Archimedes parabola area, and base height parabola segment.
People want a numeric result they can check against a worksheet key, a hull loft, or a CAD note, not only a geometric argument that the figure is determined.
What you can do with it
Homework and Archimedes quadrature drills
Students and graders confirm that a parabolic segment area is (2/3)·base·height and that the segment is (4/3) of the inscribed triangle with the same base and height.
FAQ
How do I calculate the area of a parabolic segment?
Send non-negative finite numbers base (chord length) and height (perpendicular sagitta).
What happens if base or height is negative?
A geometric dimension cannot be negative; failed validation is not billed.
Is this the same as a circular segment area?
No. A circular segment needs radius and central angle (or equivalent). A parabolic segment here uses only base and height via Archimedes’ two-thirds formula.
Why is the area (2/3)·base·height?
Archimedes proved the parabolic segment is four-thirds the area of the inscribed triangle with the same base and height. That triangle has area (1/2)bh, so the segment is (4/3)·(1/2)bh = (2/3)bh.
Where exactly is the height of a parabolic segment measured?
From the midpoint of the base chord, straight out to the curve at a right angle to the chord. This distance is also called the sagitta, and it uses the same length unit as the base.
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/parabolic-segment-area \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"base":6,"height":3}'const res = await fetch("https://api.kit.forhosting.com/geom/parabolic-segment-area", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"base": 6,
"height": 3
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/parabolic-segment-area",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"base": 6,
"height": 3
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/parabolic-segment-area", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"base":6,"height":3}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"base":6,"height":3}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/parabolic-segment-area", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"base": 6,
"height": 3
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.parabolic_segment_area",
"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 parabolic segment is the region bounded by an arc of a parabola and the straight chord that joins the arc’s endpoints. This capacity accepts two non-negative finite numbers—base and height—rejects any negative dimension with a clear invalid_input error, and returns the segment area together with the inscribed-triangle area and a short formula string as structured JSON.
What a parabolic segment is and why Archimedes’ formula fixes its area
Delivering area, triangle_area, formula, and method as JSON makes the result easy to paste into notes, diff in automated tests, or feed into a larger pipeline without reimplementing the two-thirds identity in every language you teach or deploy.
How this calculator validates input and applies the two-thirds identity
Edge cases are explicit: missing fields, empty strings, non-numeric values, and any strictly negative dimension all fail before any area is returned. Units of length are whatever you already use for the figure; response lengths and areas share that same unit system without conversion. Send a JSON body with two required numeric fields: base and height. Each must be a finite number (JSON number or numeric string). Dimensions must be non-negative: a negative base or a negative height fails immediately with invalid_input and a message that names the offending field and the non-negative rule, so callers never receive a silent absolute-value wrap or a negative area from a sign error. Results are rounded to a stable decimal precision so the same input always yields the same JSON across runtimes.
Practical use, limits, pricing, and delivery channels
Use the browser tool when you want an immediate parabolic segment area while checking homework on Archimedes’ quadrature, sizing a parabolic arch cut-out, verifying a reflector cross-section, or confirming a shop drawing without installing a computer-algebra system. Accepted dimensions stay non-negative and finite; the calculator does not invert for unknown base or height given only the area, does not fit a parabola from scattered points, and does not compute circular segments—those are different search intents and different formulas. 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. Results are fully deterministic: the same base and height always yield the same area, triangle_area, formula, and method fields. Wire the endpoint into history-of-math drills, contest answer checkers, arch and hull worksheets, or floating preprocessing steps where you already prefer explicit domain errors over opaque black-box widgets that cannot show why a negative sagitta is rejected.
Integration use cases
Deterministic fixtures for geometry tests
Pipelines that seed unit tests or interactive parabola demos can call a pure parabolic-segment oracle instead of reimplementing the two-thirds identity in every language.
Arch, reflector, and cut-out checks
Field or shop notes that record a base chord and sagitta recover segment area as structured JSON for reports, lofting helpers, or lightweight CAD preprocessing.
Technical questions
What does the parabolic segment area calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.