Conic eccentricity calculator
Find how stretched a conic is: an ellipse with semi-axes 5 and 4 has an eccentricity of 0.6 and its focus 3 from the center. Circles give 0, parabolas 1 and hyperbolas more than 1. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Conic type
- Ellipse
- A
- 5
- B
- 4
- Linear eccentricity
- 3
- Eccentricity
- 0.6
Eccentricity is the single dimensionless number that tells you which conic section you have and how stretched it is. A circle has eccentricity zero, an ellipse lies strictly between zero and one, a parabola sits exactly at one, and a hyperbola always exceeds one. This capability takes the conic type plus, when needed, the semi-axes a and b, then returns the eccentricity with the formula used and—for ellipse and hyperbola—the linear eccentricity to the focus.
What conic eccentricity measures
In classical geometry a conic section is the locus of points whose distance to a fixed focus is a constant multiple of the distance to a fixed directrix. That constant multiple is the eccentricity e.
The same number appears in the standard axis forms: for an ellipse with semi-major axis a and semi-minor axis b you get e equals the square root of one minus b squared over a squared; for a hyperbola with semi-transverse a and semi-conjugate b you get e equals the square root of one plus b squared over a squared; a circle is the limiting ellipse with e equals zero; a parabola has e equals one by definition.
Because e is dimensionless, two ellipses that share the same eccentricity are similar, and likewise for hyperbolas: scaling a and b by the same factor does not change e. Students and engineers use eccentricity to compare shapes, classify a fitted curve, or connect orbital-style language (how elongated an orbit is) to plane geometry. Mixing the ellipse formula with the hyperbola formula, swapping a and b, or allowing a less than or equal to b on an ellipse are common hand-calculation mistakes.
Automating the branch by conic type with validated positive axes removes those slips while still exposing the numbers you need for the next step of a solution or drawing.
Where to use it and what it costs
Typical placements include verifying student answers that ask for eccentricity given type and axes, building catalog rows of similar ellipses or hyperbolas that share e while scaling a and b, documenting focus geometry for technical drawings, and feeding e into larger conic classifiers that already know the semi-axes from a fit. The browser path runs locally and needs no account.
Keep ellipse axes ordered with a as semi-major and b as semi-minor, keep hyperbola axes strictly positive, and prefer the server-computed eccentricity field rather than re-deriving the square root in client code that might use a different floating-point path.
What you can do with it
Homework and exam checks
Confirm a student’s eccentricity for a named conic and given axes against one canonical e, formula string, and optional linear eccentricity.
Similar-conic catalogs
Scale a and b while holding eccentricity fixed, or compute e once per shape family when building tables of ellipses and hyperbolas for teaching or design.
FAQ
What happens if ellipse axes have a less than or equal to b?
For an ellipse a must be the semi-major axis and strictly greater than the semi-minor axis b.
What eccentricity does each conic have?
A circle has 0, an ellipse lies strictly between 0 and 1, a parabola is exactly 1 and a hyperbola is always above 1. For a circle or a parabola you do not need to enter any axes.
How do I calculate the eccentricity of an ellipse?
Take the square root of one minus b squared over a squared, with a as the semi-major and b as the semi-minor axis. Semi-axes of 5 and 4 give an eccentricity of 0.6.
What is the eccentricity formula for a hyperbola?
It is the square root of one plus b squared over a squared, with a as the semi-transverse and b as the semi-conjugate axis, so the result is always greater than 1.
What is linear eccentricity?
It is the distance from the center to a focus. For an ellipse it is the square root of a squared minus b squared, 3 for semi-axes 5 and 4; dividing it by a gives the eccentricity.
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/conic-eccentricity \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"conic_type":"circle"}'const res = await fetch("https://api.kit.forhosting.com/geom/conic-eccentricity", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"conic_type": "circle"
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/conic-eccentricity",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"conic_type": "circle"
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/conic-eccentricity", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"conic_type":"circle"}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"conic_type":"circle"}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/conic-eccentricity", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"conic_type": "circle"
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.conic_eccentricity",
"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
How the capability computes the answer
For circle the engine returns eccentricity zero and does not require axes. For parabola it returns eccentricity one with the same constant path. For ellipse you must supply positive finite a and b with a strictly greater than b; the engine computes linear eccentricity as the square root of a squared minus b squared and eccentricity as the square root of one minus b squared over a squared. For hyperbola you must supply positive finite a and b; linear eccentricity is the square root of a squared plus b squared and eccentricity is the square root of one plus b squared over a squared. You send a JSON body with required field conic_type set to one of circle, ellipse, parabola, or hyperbola. Numbers may arrive as JSON numbers or plain numeric strings. Missing required fields, non-numeric values, zero or negative axes, an unknown conic type, or an ellipse with a not greater than b fail with invalid_input and are never charged. There is no randomness, no clock dependency, and no network call; two identical requests always produce the same JSON. Optional output fields such as a, b, and linear_eccentricity appear only when they apply, so circle and parabola responses stay minimal while ellipse and hyperbola responses carry everything needed to cross-check e against c divided by a.
Where to use it and what it costs
Use the free browser tool when you want a quick check for one conic, or call the API when a script, gradebook, or design job must resolve many shapes the same way every time. API usage draws from prepaid KIT balance at the published $0.002 base price per successful request; failed validation does not consume balance.
Integration use cases
CAD and focus prep
Pass measured semi-axes from a drawing into the API and store eccentricity next to linear focus distance for documentation or further geometry steps.
Technical questions
How do I calculate conic eccentricity with this API?
POST JSON with conic_type (circle, ellipse, parabola, or hyperbola). For ellipse and hyperbola also send positive a and b. The response includes eccentricity and the formula used.
Do circle and parabola need axes?
No. Circle always returns e=0 and parabola always returns e=1; a and b are not required and are not used for those types.
Is the conic eccentricity calculator free?
The browser widget is free and runs locally. API calls use prepaid KIT balance at the published $0.002 base price; invalid input is not charged.
Is the result always the same for the same inputs?
Yes. The algorithm is pure analytic math with no randomness or timestamps, so identical inputs produce identical JSON.