Parabola arc length calculator
The curve y = x² is 4.646783762433 units long from 0 to 2 along x. Enter the coefficient and the two bounds to get the exact arc length of a parabolic arch, mirror or textbook curve. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- A
- 1
- X1
- 0
- X2
- 2
- Arc length
- 4.6467837624
- Equation
- y=1x^2
Arc length along a parabola is a classic calculus exercise that still shows up in design, optics, and exam prep whenever a curve y equals a times x squared must be measured between two vertical lines. Hand integration of the square root of one plus the squared derivative is error-prone, and generic numerical quadrature is heavier than necessary for this elementary closed form. This parabola arc length calculator accepts the quadratic coefficient a and the finite bounds x1 and x2, evaluates the exact antiderivative involving a square root and an inverse hyperbolic sine, and returns a positive arc length with the echoed inputs and a short formula string.
What parabola arc length means and how the closed form works
A parabola written as y equals a times x squared is the simplest vertical quadratic. Its slope at any x is two a x, so the differential arc-length element is the square root of one plus four a squared x squared. Integrating that element from x1 to x2 yields the total curve length between the vertical lines x equals x1 and x equals x2.
When a is non-zero the antiderivative has a standard closed form: half of x times the square root of one plus b squared x squared, plus one over two b times the inverse hyperbolic sine of b x, where b equals two times the absolute value of a. Evaluating that primitive at both ends and taking the absolute difference produces a positive length regardless of whether the caller ordered the bounds ascending or descending.
The inverse hyperbolic sine form is preferred over the equivalent logarithmic expression because it stays numerically stable for large negative x. The calculator intentionally answers only this arc-length question for y equals a x squared. It does not return area under the curve, focal length, vertex coordinates, or a parametric reparameterization by arc length.
Keeping the contract to three numbers and one primary output makes the tool ideal as a fast check beside a textbook problem, a CAD dimension, or a one-line step in a larger geometry pipeline that already chose the monic quadratic form.
Where a parabola arc length calculator earns its keep
Geometry and calculus homework often asks for the length of y equals a x squared between two x-values after students have already set up the integral of the square root of one plus the squared derivative. Optical and structural design notebooks that freeze a from a focal-length relation and need the material length of a parabolic mirror or arch segment between two vertical stations want a closed-form number without importing quadrature libraries.
Landscape and architecture workflows that lay parabolic arches, ramps, or cable profiles convert a measured coefficient and two station marks into total curve length for stock cut lists without opening a full CAD suite. Exam-prep generators and LMS auto-graders need a tiny closed-form checker that rejects equal bounds as invalid rather than silently returning zero length that a buggy grader might accept.
What you can do with it
Classroom and homework checks
Students enter a, x1, and x2 for y=ax² and verify the closed-form arc length, catching chord approximations and sign mistakes on the inverse hyperbolic sine.
Mirrors, arches, and material estimates
Convert a frozen design coefficient and two station marks into total parabolic curve length for stock, cable, or mirror-edge estimates without numerical integrators.
LMS and exam auto-grading
Mark arc-length items with a deterministic oracle that rejects equal x-bounds as invalid and never invents a length for a zero-width interval.
FAQ
What happens if x1 equals x2?
Equal range bounds define a zero-width interval with no meaningful arc length and are not billed.
Does it work for y = ax² + bx + c?
No. It measures only curves of the form y = a times x squared. It also does not give the area under the curve, the focal length or the vertex; it answers the arc length question alone.
Does the order of the two bounds matter?
No. The formula is evaluated at both ends and the absolute difference is taken, so you get the same positive length whether you enter the smaller or the larger bound first.
What if the coefficient is zero?
Then the graph is a horizontal line, and the length is simply the distance between the two bounds. It is still a valid result, so a flat curve needs no special handling.
What unit is the arc length in?
The same unit you use for x. Bounds in metres give an arc length in metres and bounds in inches give inches; the calculator does not convert units for you.
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/parabola-arc-length \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"a":1,"x1":0,"x2":2}'const res = await fetch("https://api.kit.forhosting.com/geom/parabola-arc-length", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"a": 1,
"x1": 0,
"x2": 2
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/parabola-arc-length",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"a": 1,
"x1": 0,
"x2": 2
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/parabola-arc-length", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"a":1,"x1":0,"x2":2}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"a":1,"x1":0,"x2":2}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/parabola-arc-length", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"a": 1,
"x1": 0,
"x2": 2
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.parabola_arc_length",
"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
When the two bounds are equal the interval has zero width, so the task fails with invalid_input instead of reporting a vacuous zero length that a careless client might treat as success.
What parabola arc length means and how the closed form works
When a is zero the graph is a horizontal line and the length collapses to the absolute difference of the bounds, which this endpoint still reports as a valid success so callers do not need a special branch for flat coefficients.
Input contract, response shape, and error behaviour
Example: a equals one, x1 equals zero, x2 equals two yields a length near four point six five, matching the standard textbook integral for y equals x squared on that interval. Units are caller-defined: if you send metres for x you get metres of arc length; if you send inches you get inches. The calculator does not convert units, prove that a came from a physical focus parameter, draw the parabola, or switch to a general y equals a x squared plus b x plus c form. Pair the numeric result with your own unit labels and drawings. POST a JSON body with three required finite numbers: a, the quadratic coefficient in y equals a x squared; x1, the first bound of the x-interval; and x2, the second bound. Each field may arrive as a JSON number or as a numeric string; empty strings, missing fields, Infinity, and NaN fail with invalid_input naming the problem. Aliases coefficient and coeff map to a; x_min, from, and start map to x1; x_max, to, and end map to x2 so clients that already use longer geometry names do not need a rename layer. On success the body includes the echoed a, x1, and x2 rounded to twelve decimal places for stable goldens; arc_length equal to the absolute difference of the closed antiderivative at the two bounds with the same rounding; equation equal to a short y equals a x squared string; and formula restating the integral identity. Keep requests small, pin the rounded arc_length in tests, and you get reproducible three-field answers for homework, LMS plugins, and edge workers without standing up a numerical integrator.
Where a parabola arc length calculator earns its keep
Learners sometimes replace the integral by the straight-line chord between the endpoints, which underestimates the true curve; a deterministic oracle that always applies the closed form catches those shortcuts before a tutor spends a session on hyperbolic substitutions. Engineering notebooks and spreadsheet replacements can call this endpoint as a three-field step when a larger script already holds coefficients for several figures and only the pure y equals a x squared arc-length case needs a dedicated, SEO-clear name distinct from circular sector arc length or full parabola vertex suites. In all of those settings the value is not exotic mathematics: it is a stable contract, clear errors, and a search intent that matches how people actually query for parabola arc length from a coefficient and an x-range at $0.002 per successful API call.
Technical questions
How do I compute the arc length of a parabola y = a x²?
Send finite numbers a, x1, and x2. The API integrates √(1+(2ax)²) in closed form and returns the positive arc_length between those vertical bounds.
Is a = 0 allowed?
Yes. When a is zero the graph is a horizontal line and arc_length equals the absolute difference |x2−x1|, provided the bounds still differ.
Does the order of x1 and x2 matter?
No. The returned arc_length is always non-negative; the absolute difference of the antiderivative values makes either order valid.
What does the parabola arc length calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.