Solve an AAS triangle
Finish a triangle from two angles and a non-included side: angles of 50° and 60° leave 70° for the third, and the law of sines then gives both missing sides and the area, in your unit. 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
- 10
- Angle c
- 70
- Side b
- 11.3051587485
- Side c
- 12.2668159691
- Area
- 53.1168712638
This AAS triangle solver completes an oblique triangle when you know two interior angles and a non-included side. It first checks that the supplied angles can belong to a real triangle, finds the remaining angle from the 180-degree sum, and then applies the law of sines to calculate both unknown side lengths. It also returns the triangle's area, giving you a complete structured result for homework, surveying calculations, design checks, or automated geometry workflows. Every calculation is deterministic and uses no external service.
Identify and enter the AAS measurements correctly
AAS stands for angle-angle-side. You must know two interior angles and a side that is not the included side between those angles. The returned side lengths use that same unit, and the area uses its squared form.
Before accepting the measurements, confirm that your diagram really labels the supplied side opposite angle A. If your known side lies between the two known angles, your measurements describe ASA rather than AAS and should be mapped using the appropriate solver. Correct labels prevent a mathematically valid calculation from representing the wrong physical triangle.
Once the labels are aligned, one AAS data set determines exactly one triangle, so there is no ambiguous second solution to compare.
Understand the angle, side, and area calculations
Every Euclidean triangle has interior angles totaling 180 degrees. After finding a positive third angle, the calculator uses the law of sines. Rearranging those equal ratios produces both missing lengths.
Trigonometric functions receive radians internally even though the public input and output angles use degrees. The calculation does not estimate a drawing, call an optimization service, or choose among multiple geometric candidates. It follows direct formulas, making the output suitable for checking hand calculations and for deterministic tests in software that processes triangle measurements.
Interpret results and use them safely
The response includes all three angles, all three sides, the area, the angle unit, and the AAS method label. The original measurements are echoed alongside the computed values, which makes a stored response understandable without separately preserving every input field. For example, an input side measured in meters produces other sides in meters and area in square meters.
Very narrow triangles can yield extreme proportions because a small sine appears in the law-of-sines ratio, so retain enough precision when using the result downstream. If field angles or lengths have tolerances, evaluate those tolerances separately rather than treating the displayed decimal digits as guaranteed physical accuracy.
What you can do with it
Check geometry exercises
Verify a student's third angle, missing side lengths, and area from a complete AAS exercise.
Complete survey notes
Turn two recorded angles and an opposite baseline into the remaining triangle dimensions for a draft report.
FAQ
Why must the two known angles sum to less than 180 degrees?
The third interior angle is 180 minus their sum. A sum of 180 or more would leave no positive third angle, so no triangle exists.
Which units does the result use?
Area uses the corresponding square unit, while all angles are in degrees.
Is there an ambiguous case for AAS?
No. Two valid angles fix the shape, and the known opposite side fixes its scale, so AAS determines one triangle.
How do I find the third angle of an AAS triangle?
Subtract the two known angles from 180°, since the interior angles of a flat triangle always add up to 180°. With 50° and 60° known, the third angle is 70°.
What is the difference between AAS and ASA?
In AAS the known side lies opposite one of the known angles, not between them. If your known side sits between the two angles, the data describe ASA and belong in a different solver.
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/trig/solve-aas-triangle \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"angle_a":50,"angle_b":60,"side_a":10}'const res = await fetch("https://api.kit.forhosting.com/trig/solve-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": 10
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/trig/solve-aas-triangle",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"angle_a": 50,
"angle_b": 60,
"side_a": 10
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/trig/solve-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":10}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"angle_a":50,"angle_b":60,"side_a":10}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/trig/solve-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": 10
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "trig.solve_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
Identify and enter the AAS measurements correctly
This calculator names the known angles angle_a and angle_b, while side_a is the known length opposite angle_a. That opposite relationship matters because the law of sines pairs each side with the sine of the angle directly across from it. Enter both angles in degrees and enter side_a in any consistent length unit, such as centimeters, feet, or meters. Each angle must be a finite positive number below 180 degrees, while side_a must be finite and greater than zero.
Understand the angle, side, and area calculations
The solver therefore calculates angle_c by subtracting angle_a and angle_b from 180. It validates the sum first: if the two supplied angles total 180 degrees or more, the remaining angle would be zero or negative and no ordinary triangle could exist. This law says that side_a divided by sine of angle_a equals side_b divided by sine of angle_b and also equals side_c divided by sine of angle_c. The area then comes from one-half times side_b times side_c times the sine of their included angle, which is angle_a.
Interpret results and use them safely
Check that angle_a, angle_b, and angle_c total 180 degrees apart from harmless decimal rounding. Side_a remains in the unit you supplied; side_b and side_c inherit that unit, while area is expressed in square units. The API rejects missing values, non-finite numbers, non-positive lengths, invalid individual angles, and angle pairs whose sum is not strictly below 180 degrees. A successful item costs $0.002; the browser version can run the same pure calculation locally. This tool is useful for education, preliminary design, and routine computational checks, but measurement uncertainty still belongs to the user.
Integration use cases
Automate design calculations
Produce stable structured triangle values for CAD helpers, test fixtures, and engineering worksheets.
Technical questions
What measurements does the calculator require?
Provide angle_a and angle_b in degrees and side_a as the positive length opposite angle_a.
How much does an API calculation cost?
Each successful API item costs $0.002. The calculation is also available through the browser experience.