AAS triangle calculator
With angles of 50 and 60 degrees and a side of 8 opposite the first, the third angle is 70 degrees and the whole triangle is fixed. Enter two angles and one non-included side to get every angle and side at once. 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
- Side a
- 8
- Angle c
- 70
- Side b
- 9.0441269988
- Side c
- 9.8134527752
- Angle unit
- deg
AAS stands for angle-angle-side: you know two interior angles of a plane triangle and one side that is not included between them. Because interior angles always sum to one hundred eighty degrees, the third angle is immediately determined, and the law of sines then fixes the two unknown sides from the known one.
What the AAS case is and why two angles plus a non-included side fix a triangle
In Euclidean plane geometry the three interior angles of a triangle always add up to one hundred eighty degrees, so knowing any two of them fully determines the third: subtract their sum from one hundred eighty. The AAS case adds one length to that picture — a side that is not the side lying between the two known angles.
If the known side lay between them, you would have the ASA case instead; here the side is opposite one of the known angles, conventionally written as angle A, angle B, and side a. Two angles already fix the shape of the triangle up to scale, because every triangle with those angles is similar to every other. The single known length then fixes the scale, so exactly one triangle satisfies the data.
That is why AAS appears alongside ASA, SAS, and SSS as a congruence criterion in geometry courses: the data genuinely pin the triangle down. Search intent clusters around phrases like AAS triangle calculator, angle angle side solver, and find the missing side and angles from two angles. What people want is a numeric answer they can check against a textbook key or feed into a worksheet, not only a proof that such a triangle exists.
Practical use, limits, pricing, and delivery channels
Use the free browser tool when you want an immediate AAS solution while checking homework, drafting a geometric construction, verifying a survey sketch, or preparing lesson materials without installing a computer-algebra system. The known side must be a positive finite length in whatever unit your figure already uses — millimeters, meters, inches — and the returned sides share that same unit, while angles always come back in degrees.
There is no language model, no external geometry service, and no randomness anywhere in the path, which makes the capability safe to pin inside continuous-integration checks, reproducible textbooks, and automated answer graders. Typical integrations include congruence-criterion demos that contrast AAS with ASA and SSA, contest answer checkers, engineering worksheets that list two angles and an opposite side, and preprocessing steps in larger solvers where you prefer explicit angle-sum logic and the law of sines over opaque black-box widgets.
FAQ
What does AAS mean in a triangle problem?
AAS means angle-angle-side: two interior angles are known together with one side that is not included between them. The third angle follows from the 180-degree angle sum, and the remaining sides follow from the law of sines.
How is the third angle found?
Subtract the two known angles from 180 degrees. With 50 and 60 degrees, the third angle is 70 degrees, and the law of sines then gives the missing sides.
What is the difference between AAS and ASA?
In ASA the known side lies between the two known angles; in AAS it sits opposite one of them. Both give exactly one triangle, but the side you know is in a different place.
What happens if my two angles add up to 180 degrees or more?
No triangle exists, so the calculator stops with a clear message instead of returning a result. Two interior angles must leave room for a positive third angle.
What unit do the missing sides use?
The same unit as the side you enter, whether millimetres, metres or inches. Angles always come back in degrees, so a side of 8 metres returns the other two sides in metres.
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/aas-triangle \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"angle_a":50,"angle_b":60,"side_a":8}'const res = await fetch("https://api.kit.forhosting.com/math/aas-triangle", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"angle_a": 50,
"angle_b": 60,
"side_a": 8
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/aas-triangle",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"angle_a": 50,
"angle_b": 60,
"side_a": 8
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/aas-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,"side_a":8}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"angle_a":50,"angle_b":60,"side_a":8}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/aas-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,
"side_a": 8
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.aas_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
This capability takes angle_a, angle_b, and side_a (the side opposite angle_a), rejects any pair of angles whose sum reaches one hundred eighty degrees or more, and returns the complete triangle — all three angles and all three sides — as deterministic JSON.
What the AAS case is and why two angles plus a non-included side fix a triangle
Returning the full set of angles and sides as structured JSON makes the result trivial to paste into notes, diff inside automated tests, or pipe into a larger geometry pipeline without re-deriving degree-to-radian conversion and sine ratios by hand every time.
How this solver validates input, derives the third angle, and applies the law of sines
Angles are measured in degrees and must be positive interior values strictly less than one hundred eighty. The defining domain rule of the AAS case comes next: the two given angles must sum to strictly less than one hundred eighty degrees. Non-positive or non-finite sides, missing fields, and out-of-range angles fail the same way. For accepted input the third angle is computed as one hundred eighty minus the sum of the two known angles. Send a JSON object with the required fields angle_a, angle_b, and side_a. The engine first checks that each field is present and parses to a finite number, accepting both JSON numbers and numeric strings so form submissions and query parameters work without pre-processing. If angle_a plus angle_b equals or exceeds one hundred eighty, no plane triangle exists — two interior angles alone would already consume the entire angle budget — and the request fails with an invalid_input error before any trigonometry runs, so failed calls are never charged. The law of sines states that each side divided by the sine of its opposite angle is constant, so side_b equals side_a times sine of angle_b over sine of angle_a, and side_c equals side_a times sine of angle_c over sine of angle_a. The response carries every angle and every side plus an explicit angle_unit of degrees and a method tag of AAS, so downstream code can distinguish this solve from an ASA or AAA result with overlapping field names.
Practical use, limits, pricing, and delivery channels
Each successful request is one billable item at $0.002; input that fails validation does not consume balance, so exploratory calls are safe. Everything is fully deterministic: identical angle_a, angle_b, and side_a always yield identical angle_c, side_b, and side_c. If you instead know all three angles, use the AAA variant; if your known side is included between the angles, use the ASA variant — this endpoint is specifically the non-included-side case.
Integration use cases
Homework and exam AAS drills
Surveying and layout verification
Automated answer graders and CI fixtures
Technical questions
Why does the request fail when my two angles sum to 180 degrees or more?
The three interior angles of any plane triangle must sum to exactly 180 degrees. If two of them already reach 180 or more, nothing is left for the third angle, so no such triangle exists and the solver returns an invalid_input error.
Which side should I pass as side_a?
Pass the side opposite angle_a — that is what makes the case AAS rather than ASA. If your known side lies between the two known angles, use an ASA solver instead.
What units do the results use?
Angles are always returned in degrees. Sides are returned in whatever unit you used for side_a; the solver performs pure ratios, so the unit carries through unchanged.
How much does an API call cost and are failed calls charged?
Each successful request is one item at $0.002. Requests that fail validation — such as angles summing to 180 degrees or more — return invalid_input and are not charged.