Hyperbola latus rectum calculator
Get the full width of a hyperbola through its focus from the two semi-axes: 3 and 4 give a latus rectum of 2 × 4² ÷ 3, a little over 10. It is a plain length in the unit of your axes. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- A
- 3
- B
- 4
- Latus rectum
- 10.6666666667
The latus rectum of a hyperbola is the chord through either focus that is perpendicular to the transverse axis. For the standard form with semi-transverse axis a and semi-conjugate axis b, its full length is the closed formula two b squared over a. This capability accepts those two positive semi-axis lengths and returns the latus rectum length together with the inputs and the formula string, so homework checks, CAD helpers, and geometry pipelines can reuse one deterministic result instead of re-deriving the quotient by hand.
What the hyperbola latus rectum means
In conic-section geometry a hyperbola is the set of points whose absolute difference of distances to two fixed foci is constant. That constant is two a, the full transverse axis length, while the distance from the center to each focus is c, with c squared equal to a squared plus b squared. Through each focus there is a chord perpendicular to the transverse axis; that chord is the latus rectum, and its full length for a non-degenerate hyperbola is L equals two b squared over a.
The same formula applies whether the transverse axis is horizontal or vertical, as long as a is the semi-transverse axis and b is the semi-conjugate axis. Textbooks use the latus rectum to relate focal width to the opening of the curve: larger b relative to a widens the chord through the focus, while larger a stretches the transverse direction and shortens L. Students often confuse the ellipse formula two b squared over a with the hyperbola one, mix a and b, or compute only half the chord.
The result is a pure length in the same units as a and b, not an eccentricity or a slope, so it slots cleanly next to focus distance and asymptote slope in a full element list.
Where to use it and what it costs
Typical placements include verifying student answers that ask for the latus rectum given a and b, generating catalog rows of hyperbolas for teaching materials, documenting focal chord width next to foci and vertices in technical drawings, and feeding L into larger conic classifiers that already know the semi-axes from a fit. The browser path runs locally and needs no account.
If you already use a related hyperbola helper for eccentricity or full element lists, this landing targets the distinct search intent for latus rectum length alone with the same transparent pricing model.
What you can do with it
Homework and exam checks
Confirm a student’s latus rectum for given semi-axes a and b against one canonical L equals 2b²/a instead of re-grading hand arithmetic.
Conic element tables
Build teaching or design tables that list a, b, and focal chord length L for each hyperbola family without mixing ellipse and hyperbola formulas.
FAQ
What happens if a or b is negative?
Semi-axis lengths must be strictly positive geometric lengths; negative values are rejected.
What is the latus rectum of a hyperbola?
It is the chord that passes through a focus and runs perpendicular to the transverse axis. The same length holds whether that axis is horizontal or vertical, as long as a is the semi-transverse axis.
What is the formula for the hyperbola latus rectum?
Twice the semi-conjugate axis squared, divided by the semi-transverse axis. With semi-axes 3 and 4 that is 2 × 4² ÷ 3, which comes out a little over 10.
How does the latus rectum change with a and b?
A larger b relative to a widens the chord through the focus, while a larger a stretches the curve along the transverse direction and shortens the latus rectum.
What mistakes are common with the latus rectum?
Swapping a and b, computing only half of the chord, or mixing it up with the ellipse case. The answer is a length in the unit of the semi-axes, never an eccentricity or a slope.
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/hyperbola-latus-rectum \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"a":3,"b":4}'const res = await fetch("https://api.kit.forhosting.com/geom/hyperbola-latus-rectum", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"a": 3,
"b": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/hyperbola-latus-rectum",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"a": 3,
"b": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/hyperbola-latus-rectum", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"a":3,"b":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"a":3,"b":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/hyperbola-latus-rectum", 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": 3,
"b": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.hyperbola_latus_rectum",
"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
What the hyperbola latus rectum means
Automating the calculation with validated positive inputs removes those slips while still exposing a, b, and the latus rectum in one JSON object for the next step of a solution or pipeline.
How the capability computes the answer
Both must be finite and strictly positive; zero is rejected because division by a would be undefined or the conic would be degenerate, and any negative length is rejected because a semi-axis is a geometric length, not a signed coordinate. There is no randomness, no clock dependency, and no network call; two identical requests always produce the same result object. The algorithm is a handful of multiplications and one division, so it stays well inside the analytic cost model used for pricing this tier-A parse capability. You send a JSON body with two required numeric fields: a, the semi-transverse axis, and b, the semi-conjugate axis. The engine parses numbers from JSON numbers or plain numeric strings, then computes the latus rectum as two times b squared divided by a, with stable rounding so identical inputs yield identical JSON across runtimes. The response reports a, b, latus_rectum, and a short formula string L equals two b squared over a. Missing fields, non-numeric values, zero, or negative semi-axes fail with invalid_input and are never charged. Prefer the server-computed latus_rectum field rather than re-deriving the quotient in client code that might use a different floating-point path or forget the factor of two.
Where to use it and what it costs
Use the free browser tool when you want a quick check for a single pair of semi-axes, or call the API when a script, gradebook, or design job must resolve many hyperbolas 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 focal-width prep
Pass measured semi-axes from a drawing into the API and store the latus rectum next to focus distance for documentation or further geometry steps.
Technical questions
How do I calculate the hyperbola latus rectum with this API?
POST a JSON body with positive numbers a (semi-transverse) and b (semi-conjugate). The response includes latus_rectum equal to 2b²/a.
What formula does the calculator use?
For a standard hyperbola with semi-transverse axis a and semi-conjugate axis b, the full latus rectum length is L = 2b²/a.
Is the hyperbola latus rectum 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 a and b?
Yes. The algorithm is pure analytic math with no randomness or timestamps, so identical inputs produce identical JSON.