AAS triangle calculator
Solve an angle-angle-side triangle in one step: with angles of 40° and 60° and a side of 7 opposite the first, the third angle is 80° and the other sides are 9.4310744873 and 10.7246222037. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Angle a
- 40
- Angle b
- 60
- Side a
- 7
- Angle c
- 80
- Side b
- 9.4310744873
- Side c
- 10.7246222037
- Angle unit
- deg
AAS means angle-angle-side: you already know two interior angles of a plane triangle and one side that is not sandwiched between those two angles. The third angle is fixed by the fact that interior angles sum to one hundred eighty degrees, and the unknown sides follow from the law of sines once that third angle is known. Students meet this case in congruence proofs, construction problems, and exam drills that name AAS, ASA, SAS, and SSS as separate congruence patterns. Surveyors, CAD assistants, and scripted geometry checkers need a deterministic numeric answer they can pin in continuous integration rather than a drawing that only works for one figure.
What AAS is and when two angles plus a non-included side determine a triangle
In Euclidean plane geometry, the sum of the three interior angles of a triangle is always one hundred eighty degrees. If you know any two interior angles, the third is completely determined: subtract their sum from one hundred eighty. AAS then adds a length: the known side is not the side that lies between the two known angles (that would be the included side of the ASA case).
Instead the known side is opposite one of the known angles. With that information the shape is fixed up to scale by the angles, and the absolute scale is fixed by the known length through the law of sines. Classroom worksheets often phrase this as angle A, angle B, and side a, or as two angles and the side opposite the first.
Search intent clusters around phrases like AAS triangle calculator, angle angle side solver, and find the third angle and remaining sides. People want a numeric result they can check against a textbook answer key, not only a geometric argument that two triangles are congruent.
What you can do with it
Homework and exam AAS drills
Students and graders confirm the third angle and remaining sides for classic AAS data without a local CAS, including rejection when the two given angles sum to 180° or more.
FAQ
What is the difference between AAS and ASA?
AAS uses two angles and a non-included side (a side not between those two angles). ASA uses two angles and the included side between them.
What happens if the two given angles sum to 180 or more?
Two angles that sum to 180° or more cannot form a plane triangle; those calls are not charged.
Is the AAS triangle calculator free?
The browser tool runs free on your device.
How do I find the third angle of an AAS triangle?
Subtract the sum of the two known angles from 180 degrees. With angles of 40 and 60 degrees the third angle is 80 degrees, and the remaining sides then follow from the law of sines.
How are the missing sides worked out from just one side?
With the law of sines: each side divided by the sine of its opposite angle gives the same ratio. If the side of 7 faces the 40° angle, the other two come out at 9.4310744873 and 10.7246222037.
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/triangle-aas \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"angle_a":40,"angle_b":60,"side_a":7}'const res = await fetch("https://api.kit.forhosting.com/geom/triangle-aas", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"angle_a": 40,
"angle_b": 60,
"side_a": 7
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/triangle-aas",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"angle_a": 40,
"angle_b": 60,
"side_a": 7
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/triangle-aas", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"angle_a":40,"angle_b":60,"side_a":7}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"angle_a":40,"angle_b":60,"side_a":7}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/triangle-aas", 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": 40,
"angle_b": 60,
"side_a": 7
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.triangle_aas",
"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 capacity accepts angle A, angle B, and side a opposite A, rejects pairs of angles that sum to one hundred eighty degrees or more, and returns the full set of angles and sides as structured JSON.
What AAS is and when two angles plus a non-included side determine a triangle
Delivering angle C together with sides b and c as JSON makes the result easy to paste into notes, diff in automated tests, or feed into a larger pipeline without reimplementing degree-to-radian conversion and sine ratios in every language you teach.
How this calculator validates input, finds the third angle, and applies the law of sines
Angles are measured in degrees as positive interior values strictly less than one hundred eighty. The engine checks that each field is a finite number, that both angles are positive and less than one hundred eighty, and that the known side is positive. For accepted input the third angle is computed as one hundred eighty minus the sum of the two known angles. Edge cases are explicit: nearly flat angle pairs that still sum below one hundred eighty remain valid, while a sum of exactly one hundred eighty or more is rejected before any sine division. Send a JSON body with required fields angle_a, angle_b, and side_a. The critical domain rule for AAS is that the sum of the two given angles must be strictly less than one hundred eighty degrees; if the sum is one hundred eighty or more, no plane triangle exists and the request fails immediately with invalid_input so those failed calls are not charged. Missing fields, non-numeric values, non-positive sides, and out-of-range angles also fail with invalid_input. The law of sines then states that side_a over sine of angle_a equals side_b over sine of angle_b equals side_c over sine of angle_c. Solving for the unknowns gives side_b equal to side_a times sine of B over sine of A, and side_c equal to side_a times sine of C over sine of A. Results are rounded to a stable decimal precision so the same input always yields the same JSON.
Practical use, limits, pricing, and delivery channels
Use the browser tool when you want an immediate AAS solution while checking homework, drafting a construction, or verifying a survey sketch without installing a computer-algebra system. Accepted angles must form a positive third angle, and the known side must be a positive finite length in whatever unit you already use for the figure; the response sides share that same unit. There is no language model, no external geometry service, and no randomness, so the capacity is safe to pin in continuous-integration checks and reproducible lesson materials. Each successful request is one billable item at $0.002; failed validation does not consume balance. Results are fully deterministic: the same angle_a, angle_b, and side_a always yield the same angle_c, side_b, and side_c fields. Wire the endpoint into congruence demos, contest answer checkers, engineering worksheets that list AAS data, or integer and floating preprocessing steps where you already prefer exact angle-sum logic and the law of sines over opaque black-box widgets that cannot show why two angles summing to one hundred eighty degrees or more are rejected.
Integration use cases
Deterministic fixtures for geometry tests
Pipelines that seed unit tests or interactive congruence demos can call a pure AAS oracle instead of reimplementing degree trig and the law of sines in every language.
Survey and CAD side recovery from two angles
Field notes that record two angles and one opposite side can recover the missing lengths as structured JSON for import into reports or lightweight CAD helpers.
Technical questions
How do I solve an AAS triangle with this API?
Send a JSON body with angle_a and angle_b in degrees plus side_a (the side opposite angle_a) to geom.triangle_aas. The response returns angle_c and the remaining sides side_b and side_c.
Which formula finds the remaining sides?
After angle_c = 180 − angle_a − angle_b, the law of sines gives side_b = side_a · sin(angle_b) / sin(angle_a) and side_c = side_a · sin(angle_c) / sin(angle_a).