Circle from center and point
Give the center and one point on the circle and get its radius and equation: a center at the origin and the point (3, 4) give a radius of 5 and x² + y² = 25. The general form, diameter and squared radius come too. Free in your browser.
- Free
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- Your data stays on your device
Run — free
Worked example
- H
- 0
- K
- 0
- X
- 3
- Y
- 4
- Radius
- 5
- Radius squared
- 25
- Diameter
- 10
- Standard equation
- x^2 + y^2 = 25
Finding the equation of a circle when you already know the center and one point on the rim is a core analytic-geometry task in algebra courses, CAD checks, and game collision setup. Students expand (x minus h) squared plus (y minus k) squared equals r squared by hand; scripts need the same identity without a full computer-algebra stack. This circle from center and point calculator accepts the center coordinates h and k together with a point x and y on the circumference, computes the Euclidean radius as the distance from center to point, and returns the standard equation plus general form, diameter, and squared radius.
What a center and one point tell you
In the Euclidean plane a circle is completely fixed once you know its center and its radius. When the radius is not given directly, any single point that must lie on the circumference supplies that missing length: the radius is exactly the distance between the center and that point. This calculator turns that definition into a ready-to-use equation.
You provide four finite numbers: h and k for the center, and x and y for a point on the rim. Internally the engine evaluates r equals the square root of (x minus h) squared plus (y minus k) squared using a stable hypot path, then builds the standard form (x minus h) squared plus (y minus k) squared equals r squared with sign-aware terms so a negative center coordinate reads as plus rather than double minus.
Example: center at the origin and point three four yields radius five and the familiar equation x squared plus y squared equals twenty-five. Example with an offset center: h equals two, k equals negative one, point five four yields radius five and (x minus two) squared plus (y plus one) squared equals twenty-five. If you need a circle through three points without a known center, use the dedicated three-point tool; if you already know radius and only want equation forms, use the circle equation calculator.
The search intent here is specifically center plus one circumference point into radius and standard equation, or a clear error when the point is not distinct from the center.
Where this circle calculation is useful
Classrooms and tutoring platforms use a circle from center and point calculator so learners can check radius and standard form for arbitrary centers without arithmetic slips on signs. Exam-prep generators need a deterministic oracle that marks wrong the case when the given point is the center instead of printing a zero-radius equation that a buggy grader might accept. Game and simulation teams place lights, sensors, or collision circles from a known center and a boundary sample, then want the equation string or radius for debug overlays.
Failed validation does not consume balance. Keep requests small, pin the rounded radius and equation strings in tests, and you get reproducible circle data for homework, LMS plugins, and edge workers without standing up symbolic math infrastructure of your own.
What you can do with it
Homework and exam checks
Enter center and a circumference point to verify radius and the standard equation (x−h)²+(y−k)²=r², including rejection when the point equals the center.
CAD, layout, and game circles
Build a circle from a known center and one snap or boundary point for clearances, cutouts, lights, or collision radii in tools and simulations.
FAQ
How do I find the circle equation from center and a point?
Send finite numbers h, k for the center and x, y for a point on the rim.
What happens if the point equals the center?
A point on the center would force radius zero, which is not treated as a valid circle here, and invalid input is not billed.
How is the radius calculated from the center and a point?
It is the straight distance between them: the square root of the horizontal gap squared plus the vertical gap squared. A center at the origin and the point (3, 4) give a radius of exactly 5.
Does it also give the general form of the equation?
Yes. It expands the standard form into x² + y² + Dx + Ey + F set equal to zero, where D is minus twice h, E is minus twice k and F is h² + k² − r², and lists D, E and F separately.
How are negative center coordinates written in the equation?
With plus signs, the way a textbook writes them: a center at a negative x appears as (x + a)², not with a double minus. You can copy the equation into homework without fixing the signs.
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/circle-from-center-point \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"h":0,"k":0,"x":3,"y":4}'const res = await fetch("https://api.kit.forhosting.com/geom/circle-from-center-point", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"h": 0,
"k": 0,
"x": 3,
"y": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/circle-from-center-point",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"h": 0,
"k": 0,
"x": 3,
"y": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/circle-from-center-point", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"h":0,"k":0,"x":3,"y":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"h":0,"k":0,"x":3,"y":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/circle-from-center-point", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"h": 0,
"k": 0,
"x": 3,
"y": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.circle_from_center_point",
"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
If the point coincides with the center the radius would be zero, which does not describe a circle, so the tool fails with invalid_input instead of inventing a degenerate result.
What center plus a rim point determines and what the response contains
The response echoes the four inputs, returns center and point objects, radius, radius_squared, diameter, standard_equation, general_equation of the form x squared plus y squared plus D x plus E y plus F equals zero, the general coefficients D E F, and a short formula string restating the distance identity. Missing fields, non-finite values, and out-of-range magnitudes raise invalid_input naming the field.
Input contract, error when point equals center, and how equations are written
Absolute values larger than one trillion are rejected so accidental unit mistakes do not produce absurd radii. After parsing, the engine computes the Euclidean distance from center to point. On success, numbers are rounded to twelve decimal places for stable goldens across runtimes, while equation strings use compact formatting without trailing noise. Standard form prefers x squared when h is zero and (x plus a) squared when h is negative so students can copy the string into homework without fixing double signs. General form expands to D equals negative two h, E equals negative two k, and F equals h squared plus k squared minus r squared, which is useful when a downstream tool expects the monic expanded polynomial. POST a JSON body with four required finite numbers: h (aliases cx, center_x), k (aliases cy, center_y), x (aliases px, point_x), and y (aliases py, point_y). Each field may arrive as a JSON number or as a decimal string suitable for forms and spreadsheets; empty strings, Infinity, NaN, and non-numeric text fail with invalid_input. Pair the free browser run on this page with the priced API when you need the identical mathematical result inside an automated pipeline without re-implementing distance and expansion yourself. Near-zero but nonzero distances still produce a valid tiny circle; only exact coincidence of center and point is rejected as invalid_input.
Where a circle-from-center-and-point API earns its keep
Whether you arrived searching for the radius from center and point or for the standard equation of a circle given center and a rim point, the contract is the same: four coordinates in, a positive radius and readable equations out, or a clear invalid_input when the point is not distinct from the center.
Integration use cases
Deterministic geometry fixtures
Generate reproducible radius and equation strings for unit tests, contest validators, and CI without hand-rolled distance and expansion code.
Technical questions
Which equation forms are returned?
standard_equation in center-radius form and general_equation as x²+y²+Dx+Ey+F=0, plus coefficients D, E, F, diameter, and radius_squared.
Can I use aliases like cx and point_x?
Yes. Center accepts h, cx, or center_x and k, cy, or center_y. The rim point accepts x, px, or point_x and y, py, or point_y.
What does the circle from center and point calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.