Oblique triangle calculator
With sides 8 and 6 and a 40° angle between them, the calculator finds the third side and the other two angles and reports an obtuse triangle. Give any three pieces, at least one a side, and get the rest plus area and perimeter. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Side a
- 8
- Side b
- 6
- Angle c
- 40
- Side c
- 5.1439025516
- Angle a
- 91.4298471497
- Angle b
- 48.5701528503
- Angle sum
- 180
An oblique triangle is any triangle that is not a right triangle: every interior angle is acute, or exactly one angle is obtuse, and none of the corners is a right angle. Surveyors, sailors, carpenters, and students meet this figure whenever they already know a mix of sides and angles and still need the rest of the figure. This oblique triangle calculator accepts the sides and interior angles you already have, classifies the data as SSS, SAS, ASA, AAS, or the ambiguous SSA case, applies the law of sines or the law of cosines, and returns the missing sides and angles together with area, perimeter, and the case it used.
Law of sines, law of cosines, and the five standard cases
An oblique triangle is solved from a mix of known sides and known angles, never from a right-angle shortcut and the Pythagorean theorem. Two identities cover every non-degenerate case. The law of sines says that each side divided by the sine of its opposite interior angle is the same constant: a over sin A equals b over sin B equals c over sin C.
Use it when you already know an angle and its opposite side, or when two angles are known so the third is one hundred eighty degrees minus their sum and every opposite pair is then available. The law of cosines says c squared equals a squared plus b squared minus two a b cosine C, and the same cyclic form for the other sides. Use it when you know all three sides (SSS) or two sides and the included angle between them (SAS).
ASA is two angles and the side that sits between them; AAS is two angles and a side opposite one of them; SSA is two sides and an angle opposite one of those sides, which can produce zero, one, or two triangles. The response lists every valid solution, the case name, and the identity used so a student can check the same arithmetic by hand without guessing which formula applies.
Input contract, units, and when the numbers cannot form a triangle
You must provide at least three independent pieces of data and at least one side — three angles alone determine shape but not size, so AAA is rejected. When two angles arrive, the third is filled as one hundred eighty minus their sum; if that remainder is not a positive angle the figure cannot close. When three sides arrive they must satisfy the strict triangle inequality: each side shorter than the sum of the other two.
When an extra field is supplied beyond the minimum, the solver computes the triangle from a sufficient subset and then checks the extra value; a mismatch means the given sides and angles cannot form a valid triangle. SSA is the delicate case: if angle A is acute and side a is shorter than the altitude b times sin A, there is no triangle; if a equals that altitude there is one right triangle; if a sits strictly between the altitude and side b there are two triangles, both returned.
An obtuse given angle requires its opposite side to be strictly longer than the other given side or the task fails.
Where an oblique triangle calculator earns its keep
Classrooms and tutoring platforms use an oblique triangle calculator so learners can check SSS, SAS, ASA, AAS, and SSA homework without mixing the two identities or silently dropping the second SSA solution. Exam generators need a deterministic oracle that rejects an impossible figure instead of returning a not-a-number that a buggy grader might treat as a pass. Navigators and outdoor instructors use the same identities for a sighted angle and two sides on a chart when no right angle is available.
Failed validation does not consume balance. Keep requests small, pin the rounded sides and angles in tests, and you get reproducible solutions for LMS plugins, field scripts, and edge workers without standing up symbolic math infrastructure of your own.
What you can do with it
Classroom and homework checks
Let students submit any mix of sides and angles and verify the remaining sides, angles, area, and the SSS, SAS, ASA, AAS, or SSA case against textbook methods.
Exam and LMS auto-grading
Mark oblique-triangle items with a deterministic oracle that rejects figures which cannot form a valid triangle and never drops the second SSA solution.
Survey and navigation closures
Close a taped traverse or a sighted bearing from two sides and an included or opposite angle without importing a full computer-algebra library.
FAQ
When does it use the law of sines versus the law of cosines?
SSS and SAS use the law of cosines. ASA, AAS, and SSA use the law of sines (after completing the third angle when two angles are known). The method field in the response names the identity.
What happens in the SSA ambiguous case?
If the given angle is acute and the opposite side sits strictly between the altitude and the other given side, two triangles exist.
Can I solve a triangle from its three angles alone?
No. Three angles fix the shape but not the size, so endless triangles match them. Give at least one side along with two other pieces, and the calculator can work out everything else.
What do SSS, SAS, ASA and AAS mean?
They say which pieces you know: three sides; two sides and the angle between them; two angles and the side between them; or two angles and a side opposite one of them. The calculator detects the case from what you enter.
What happens if the sides cannot form a triangle?
The calculation stops. Each side must be shorter than the other two added together, and two given angles must leave a positive third angle; otherwise the figure cannot close and no triangle is invented.
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/oblique-triangle \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"side_a":8,"side_b":6,"angle_c":40}'const res = await fetch("https://api.kit.forhosting.com/math/oblique-triangle", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"side_a": 8,
"side_b": 6,
"angle_c": 40
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/oblique-triangle",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"side_a": 8,
"side_b": 6,
"angle_c": 40
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/oblique-triangle", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"side_a":8,"side_b":6,"angle_c":40}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"side_a":8,"side_b":6,"angle_c":40}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/oblique-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
{
"side_a": 8,
"side_b": 6,
"angle_c": 40
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.oblique_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
If the numbers cannot form a valid triangle the task fails with invalid_input instead of inventing a figure. The same deterministic module runs free in the browser and on the API, so homework checks and automation never disagree on a borderline SSA altitude or a failed triangle inequality.
Law of sines, law of cosines, and the five standard cases
This calculator inspects which of side_a, side_b, side_c, angle_a, angle_b, and angle_c you sent, classifies the figure as SSS, SAS, ASA, AAS, or the ambiguous SSA case, and applies the matching identity.
Input contract, units, and when the numbers cannot form a triangle
Send a JSON object with any combination of the six named fields: side_a, side_b, and side_c are positive lengths in a single consistent unit; angle_a, angle_b, and angle_c are interior angles in degrees, each strictly between zero and one hundred eighty. Aliases a, b, c, A, B, and C are accepted. Degenerate, non-finite, missing, and contradictory inputs fail with invalid_input and are not billed. Successful calls return the three sides, three angles in degrees, angle_sum, area, perimeter, triangle_type (acute, obtuse, or right), the case, the method, and a short formula string.
Where an oblique triangle calculator earns its keep
Surveyors and site engineers still reduce taped distances and measured interior angles to a closed triangle before they stake a corner; calling a tiny parse endpoint avoids opening a full computer-algebra notebook for six scalars.
Technical questions
What is an oblique triangle and what does this calculator return?
An oblique triangle has no right angle. Send a sufficient mix of side_a, side_b, side_c and angle_a, angle_b, angle_c in degrees. The API returns the remaining sides and angles, area, perimeter, the case (SSS, SAS, ASA, AAS, or SSA), and the identity it used.
When do the given sides and angles fail?
The task fails with invalid_input when the sides break the triangle inequality, the angles do not leave a positive third angle that sums to 180°, SSA has no completion, only AAA is given, or extra fields contradict the solved figure.
What does the oblique triangle calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.