Circle equation calculator
Turn a center and radius into the circle's standard and general equations, or go back from the general form: radius 5 at the origin gives x² + y² = 25, an area of 25π and a circumference of 10π. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- H
- 0
- K
- 0
- Resistance (Ω)
- 5
- R squared
- 25
- Diameter
- 10
- Area
- 78.5398163397
- Circumference
- 31.4159265359
A circle equation calculator turns geometry inputs you already have into the two algebraic forms students and engineers actually write down. Give a center and a positive radius, or paste the general form x squared plus y squared plus Dx plus Ey plus F equals zero, and you get the standard equation centered at h and k, the expanded general coefficients, plus area and circumference from the same recovered radius. Homework checks, CAD notes, game collision setup, and interview prep all need those four results side by side without opening a computer-algebra system.
What the standard and general circle equations mean
The standard form of a circle is written as the square of x minus h plus the square of y minus k equals r squared. Here h and k are the center coordinates and r is the positive radius. Expanding that identity produces the general form x squared plus y squared plus Dx plus Ey plus F equals zero, where D equals negative two h, E equals negative two k, and F equals h squared plus k squared minus r squared.
Going the other way is just as mechanical: from general coefficients, the center is h equals negative D over two and k equals negative E over two, and the radius is the positive square root of h squared plus k squared minus F. If that radicand is zero or negative, the equation is a point or no real circle at all, and this tool rejects the input instead of inventing an imaginary radius.
Area is pi times r squared and circumference is two pi r—the same formulas you used in school—so a unit circle at the origin yields area pi and circumference two pi, while a circle of radius five centered at the origin yields area twenty-five pi and circumference ten pi.
Knowing both algebraic forms matters because textbooks, contest problems, and grading rubrics switch freely between them: one exercise may ask you to complete the square from general form, and the next may ask you to expand standard form into coefficients for a CAS or a graphics shader.
Practical tips, limits, and related searches
Always reject zero or negative radius in homework scripts rather than clamping, because area and circumference lose meaning for non-circles and silent clamps hide algebra mistakes. This landing targets people searching for a circle equation calculator, standard form of a circle, or general form of a circle converter; it is not a three-point circle fit or a line-circle intersection solver. Prefer center-plus-radius when you measured or were given the center and radius; prefer general form when a textbook or CAS dumped expanded coefficients.
Round-trip checks are easy: after solving from general form, re-expand D, E, and F from h, k, and r and confirm they match within floating tolerance, and confirm that area over pi equals r squared.
What you can do with it
Homework checks for standard and general circle form
Students enter center and radius or expanded coefficients and compare hand-written standard form, general form, area, and circumference against a deterministic result.
CAD notes and construction takeoff from known centers
Convert a surveyed center and radius into both algebraic forms and closed-form area and circumference for specs, BOMs, or quantity takeoff sheets.
Game and graphics setup for circular colliders
Recover h, k, r from general-form shader or physics coefficients, or expand a designer-facing center-radius pair into equation strings for debug overlays.
FAQ
What is the standard form of a circle?
The standard form is (x − h)² + (y − k)² = r², where (h, k) is the center and r is the positive radius.
What is the general form of a circle?
Center is h = −D/2, k = −E/2, and r = sqrt(h² + k² − F) when that radicand is positive.
Why is a non-positive radius rejected?
A circle needs r > 0 for a positive area and circumference. Zero describes a point; negative radius is not a real circle.
Can I send only the radius without a center?
Yes. If r is provided and h and k are omitted, the center defaults to (0, 0), which is the unit-origin case many textbooks use first.
Is the circle equation calculator free?
The browser tool is free and runs locally.
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-equation \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"h":0,"cx":-1000000000000}'const res = await fetch("https://api.kit.forhosting.com/geom/circle-equation", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"h": 0,
"cx": -1000000000000
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/circle-equation",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"h": 0,
"cx": -1000000000000
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/circle-equation", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"h":0,"cx":-1000000000000}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"h":0,"cx":-1000000000000}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/circle-equation", 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,
"cx": -1000000000000
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.circle_equation",
"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.
Limits
max_abs | 1000000000000 |
min_radius | 1e-15 |
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
The solve path is deterministic pure arithmetic—no network, no random seeds, no wall-clock dependence—so the free browser tool and the prepaid API return the same JSON after fixed decimal rounding every time you call it.
How to call the API and what comes back
When r is present, center-plus-radius wins even if general coefficients are also sent. Send a JSON body in one of two modes. Center-plus-radius mode uses field r for the positive radius (alias radius) and optional fields h and k for the center; aliases cx, center_x, cy, and center_y are accepted, and omitted center coordinates default to zero so a bare radius describes a circle at the origin. General-form mode omits r and sends d, e, and f for the coefficients D, E, and F in x squared plus y squared plus Dx plus Ey plus F equals zero; aliases D, E, F and coeff_d, coeff_e, coeff_f map to the same fields. On success the response echoes h, k, and r, adds r_squared, diameter, area, circumference, a nested center object, a nested general object with D, E, and F, human-readable standard_equation and general_equation strings, method set to center_radius or general_form, and a short formula block documenting the closed forms. Failed validation—missing both modes, a non-finite number, a non-positive radius, coefficients that yield a non-positive radius, or values outside published absolute limits—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item and zero marginal unit price.
Practical tips, limits, and related searches
Keep coordinates, coefficients, and the radius inside the published absolute range up to one trillion so intermediate products stay finite in IEEE double precision on the Worker and in the browser. Units are whatever you pass—metres, pixels, or abstract game units—as long as you stay consistent; the API does not convert units for you. When teaching, pair examples: radius five at the origin for the classic standard form x squared plus y squared equals twenty-five; center three four with radius five for a translated circle; and general form coefficients that expand back to the same figures so students can complete the square by hand and match the API line by line. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when learners type impossible coefficients.