AAA triangle calculator
Angles of 50, 60 and 70 degrees with a side of 8 opposite the 50° angle give the other sides at about 9 and just under 10. Enter all three angles and one side; if the angles miss 180, you see their real sum. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Angle a
- 50
- Angle b
- 60
- Angle c
- 70
- Side a
- 8
- Side b
- 9.0441269988
- Side c
- 9.8134527752
- Angle unit
- deg
The AAA triangle calculator takes the three interior angles of a triangle and one known side length, and returns the lengths of the other two sides. Three angles alone determine the shape of a triangle but not its size — an infinite family of similar triangles shares the same angles — so the single side supplies the missing scale. The solver first checks that your three angles actually sum to 180 degrees and rejects the input with a clear error when they do not, then applies the law of sines to compute each unknown side with stable floating-point precision.
Why three angles are not enough on their own
In geometry class you learn that two triangles with the same three angles are similar: they have identical shapes but can be any size. A 30-60-90 set square and a 30-60-90 road sign obey exactly the same trigonometry; only their scale differs. That is why the AAA triangle calculator asks for one side length in addition to the three angles.
The angles pin down every proportion of the triangle, and the known side converts those proportions into absolute lengths. Without it, the question 'how long are the other sides?' has infinitely many correct answers.
With it, there is exactly one. This is also what distinguishes AAA from the ASA and AAS cases: in those, the third angle is derived from two given angles, while here you supply all three yourself — which makes validation your responsibility, and this tool's first job.
If you are measuring a roof pitch, a parcel boundary or a camera's field of view, you will often know all three angles from a protractor or a design drawing before you know any side, so the AAA case comes up constantly in practice.
How the law of sines produces the sides
Once the angles are validated, the computation is a direct application of the law of sines: in any triangle, each side divided by the sine of its opposite angle yields the same ratio. Given side a opposite angle A, that shared ratio is a / sin(A). Every other side is then that ratio multiplied by the sine of its own opposite angle, so b = a · sin(B) / sin(A) and c = a · sin(C) / sin(A).
Angles are accepted and returned in degrees, because that is how drawings, protractors and surveying instruments report them. Each angle must be strictly between 0 and 180 degrees, and the known side must be a positive, finite number; degenerate or impossible inputs are rejected before any trigonometry runs, so you never receive a plausible-looking answer computed from a triangle that cannot exist.
Validation: the angle sum is a hard gate
The defining feature of this capability is that it refuses inconsistent input. A small numerical tolerance absorbs floating-point representation noise, but anything a human would consider wrong is rejected. This matters more than it sounds: silently 'fixing' bad angles by rescaling them would produce sides for a triangle you never asked about, and in construction, machining or navigation that kind of quiet correction is how expensive mistakes ship.
What you can do with it
Scale a surveyed triangle
A surveyor measures all three angles of a land parcel and one accessible side; the remaining boundary lengths come back in one call.
Validate design drawings
Check that the three angles on a CAD drawing actually sum to 180 degrees before the part is machined — bad input is rejected, not silently repaired.
Teach trigonometry with reproducible answers
Students verify law-of-sines homework against a deterministic solver whose output is stable to ten decimal places.
FAQ
What does it cost?
The same solver runs free and unlimited in your browser on this page.
Why do I need a side if I already have all three angles?
Three angles fix the shape of a triangle but not its size — infinitely many similar triangles share them. One side length supplies the scale so the other two sides can be computed.
Which side should I provide?
The returned sides share the same unit as the input side.
Are angles in degrees or radians?
Degrees. Each must be greater than 0 and less than 180, and the three together must equal 180.
How accurate are the results?
Sides are computed with double-precision law of sines and rounded to ten decimal places, so repeated calls and different machines return identical values.
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/math/aaa-triangle \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"angle_a":50,"angle_b":60,"angle_c":70,"side_a":8}'const res = await fetch("https://api.kit.forhosting.com/math/aaa-triangle", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"angle_a": 50,
"angle_b": 60,
"angle_c": 70,
"side_a": 8
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/aaa-triangle",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"angle_a": 50,
"angle_b": 60,
"angle_c": 70,
"side_a": 8
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/aaa-triangle", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"angle_a":50,"angle_b":60,"angle_c":70,"side_a":8}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"angle_a":50,"angle_b":60,"angle_c":70,"side_a":8}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/aaa-triangle", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"angle_a": 50,
"angle_b": 60,
"angle_c": 70,
"side_a": 8
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.aaa_triangle",
"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
It runs entirely in your browser for free, or as a deterministic API call.
Validation: the angle sum is a hard gate
If the three angles you supply sum to anything other than 180 degrees — 179.9 from a rounded measurement, 181 from a typo, or 150 from an arithmetic slip — the request fails with an explicit error naming the actual sum, and you are not charged for it. The error response uses the standard invalid_input code, so automated pipelines can catch it programmatically and route the job for correction. For API use the call costs $0.002 per request; in the browser it is free and unlimited, and both paths execute exactly the same deterministic code, so what you test by hand is what your integration will receive in production.
Technical questions
What happens if my angles do not sum to 180 degrees?
The request fails with an invalid_input error that reports the actual sum. No result is produced and the call is not billed.