Triangle circumcircle area
Enter a triangle's three sides and get the area of the circle that passes through its corners: sides 3, 4 and 5 give a circumradius of 2.5 and a circle area of 6.25π. No angles or coordinates needed. Free in your browser.
- Free
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Run — free
Worked example
- A
- 3
- B
- 4
- C
- 5
- Perimeter
- 12
- Semiperimeter
- 6
- Triangle area
- 6
- Circumradius
- 2.5
The circumcircle of a triangle is the unique circle that passes through all three vertices. Its area depends only on the three side lengths: first form the triangle area with Heron's formula, then the circumradius R equals the product of the three sides divided by four times that area, and the circle area is pi times R squared. Surveyors, CAD and game pipelines, and students working SSS problems often need that circle area without placing the triangle in a coordinate plane or measuring angles. No network calls, no random seeds, and no dependence on wall-clock time—only finite arithmetic on three scalars.
What the circumcircle area is and how the formula works
Every non-degenerate triangle has exactly one circumscribed circle: the circle that passes through all three vertices. The center of that circle is the circumcenter, and the radius R is the common distance from that center to each vertex. When you already know the three side lengths a, b, and c—and you care about the area of that circle rather than where its center sits in a plane—you can avoid coordinates entirely.
First form the semiperimeter s equals (a plus b plus c) divided by two. The triangle area K is the positive square root of s times (s minus a) times (s minus b) times (s minus c), which is Heron's formula. The classic extended law of sines then gives R equals a b c divided by four K.
The area of the circumcircle is simply pi times R squared. A right triangle is an easy sanity check: for sides three, four, and five the hypotenuse is a diameter, so R equals two point five and the circle area is pi times six point two five. Every step is elementary arithmetic plus two square roots on scalars, so there is no iteration, no sampling, and no need for angles or vertex coordinates in the input contract.
What you can do with it
Enclosing-circle area from three measured edges
Turn taped or paced edge lengths of a triangular plot or part into the area of the unique circle through the three vertices without placing the figure in coordinates.
Game, CAD, and mesh tooling around triangular faces
Size a collision, spawn, or clearance circle from three edge lengths when only SSS data is available in the scene graph or mesh.
FAQ
How do you calculate circumcircle area from three sides?
Compute triangle area K with Heron's formula, then R = a·b·c/(4K), then the circumcircle area is πR².
What is the triangle inequality?
Each side must be strictly shorter than the sum of the other two: a+b>c, a+c>b, and b+c>a.
Why are collinear sides rejected?
When a+b=c the figure is a line segment, not a triangle.
Does this need angles or vertex coordinates?
No. Only the three side lengths are required. For circumcenter coordinates from vertices, use a dedicated circumcenter calculator instead.
Is the triangle circumcircle area 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/circumcircle-area \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"a":3,"b":4,"c":5}'const res = await fetch("https://api.kit.forhosting.com/geom/circumcircle-area", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"a": 3,
"b": 4,
"c": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/circumcircle-area",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"a": 3,
"b": 4,
"c": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/circumcircle-area", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"a":3,"b":4,"c":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"a":3,"b":4,"c":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/circumcircle-area", 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,
"c": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.circumcircle_area",
"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
This capability accepts three positive side lengths, rejects triples that violate the strict triangle inequality, and returns the triangle area, the circumradius, and the circumcircle area as deterministic JSON. The same pure arithmetic powers the free browser tool and the prepaid API path, so homework checks and production scripts always agree.
What the circumcircle area is and how the formula works
An equilateral triangle with side length two has a known closed form for R and for the circle area, which you can cross-check against this endpoint. Degenerate triples where one side equals or exceeds the sum of the other two never form a triangle, so K would be zero and R would blow up; this tool rejects them with invalid_input instead of returning infinity or NaN.
How to call the API and what comes back
Use it from geometry homework pipelines, construction takeoff scripts that need enclosing-circle area from three measured edges, game-level tooling that sizes a collision or spawn circle around a triangular region, interview-prep sandboxes, or any workflow that needs a reproducible circumcircle area without installing a computer-algebra package on every machine. Send a JSON body with three fields. Field a is the first positive side length; aliases side_a and side1 are accepted. Field b is the second positive side; aliases side_b and side2 map to the same meaning. Field c is the third positive side; aliases side_c and side3 are accepted. On success the response echoes a, b, and c, adds perimeter and semiperimeter, triangle_area K from Heron, circumradius R, circumcircle_area as pi R squared, method set to heron_circumradius, and a short formula block documenting each closed form. Failed validation—missing a required field, a non-finite number, a non-positive side, a side outside published limits, or a triple that breaks the strict triangle inequality—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item. Pair this endpoint with a separate circumcenter tool when you need the center coordinates from vertex positions; this landing is deliberately about area from sides alone.
Practical tips, limits, and related searches
When teaching, prefer paired examples: three-four-five for a right triangle with R equals two point five and circle area pi times six point two five; five-five-six or an equilateral triple for a non-right case; and one-two-ten to demonstrate the hard rejection path that protects callers from a non-triangle. Prefer this tool when you measured or were given all three edges and need the area of the unique circle through the vertices. Keep side lengths inside the published positive 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, feet, or game units—as long as all three sides share the same unit; the API does not convert units for you, and the returned areas are in square units of that same length unit. Because the formula only needs sides, students can recompute s, K, R, and pi R squared by hand for small integers and match the API line by line. This landing targets people searching for a triangle circumcircle area calculator or a circumscribed circle area from three sides tool; it is not a coordinate circumcenter solver and not a pure triangle-area-only Heron page, though it reuses Heron internally. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when users type impossible triples while learning the contract. Round-trip checks are easy: after solving, confirm that perimeter equals a plus b plus c, that triangle_area squared equals s times each (s minus side) within floating tolerance, and that circumcircle_area equals pi times circumradius squared within the same tolerance.
Integration use cases
Homework and exam checks for circumcircle area
Give students three side lengths; compare their hand-computed Heron area, circumradius abc/(4K), and πR² against a deterministic API or free browser result.