Law of sines missing side calculator
Get the one length your triangle is missing: with side a of 7, angle A of 40° and angle B of 60°, side b follows from b = a × sin B ÷ sin A, and the third angle is 80°. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Side a
- 7
- Angle a
- 40
- Angle b
- 60
- Side b
- 9.4310744873
- Angle c
- 80
- Ratio
- 10.890066788
- Angle unit
- deg
When a triangle problem hands you one side, the angle opposite that side, and one more interior angle, the law of sines finds the missing length with a single sine ratio—no full congruence case analysis required. Students, tutors, and contest writers meet this AAS or ASA-style pattern constantly: worksheets label angle A, side a, and angle B, then ask for side b. Survey notes and CAD helpers record two angles and one opposite length and need a numeric side they can pin in a report.
What the law of sines says about a missing side from two angles and one side
In Euclidean plane geometry the law of sines states that each side divided by the sine of its opposite interior angle is the same constant for a non-degenerate triangle: a over sin A equals b over sin B equals c over sin C. Once you know one angle-side pair, that common ratio is fixed, so a second interior angle immediately determines the side opposite that second angle through simple multiplication and division—no need to reconstruct the whole figure first if all you want is that length.
Search intent clusters around phrases like law of sines missing side calculator, find side b with law of sines, AAS triangle side, and ASA side opposite known angle. People want a numeric length they can check against a homework answer key, a field notebook, or a dimension callout, not only a proof that two triangles are congruent.
What you can do with it
Homework AAS and ASA missing-side drills
Students and graders confirm side b from a known side a, angle A, and angle B, with clear rejection when the two angles cannot form a plane triangle.
FAQ
What formula does the missing-side calculator apply?
The law of sines: a/sin(A) = b/sin(B). Rearranged for the unknown side: b = a · sin(B) / sin(A), with angles in degrees converted to radians for the sine calls.
What happens if the two angles sum to 180° or more?
No plane triangle exists when two interior angles already fill or exceed the one-hundred-eighty-degree sum; those calls are not charged.
Is the law of sines missing side calculator free?
The browser tool runs free on your device.
Can I use this when I know two sides and need an angle?
No. This landing is side-focused: two angles and one opposite side.
How do I find side b if I know side a, angle A and angle B?
Multiply side a by the sine of B and divide by the sine of A. With 7, 40° and 60°, the calculator returns side b along with the third angle, 80°, and the shared ratio of side to sine.
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/law-of-sines-side \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"side_a":7,"angle_a":40,"angle_b":60}'const res = await fetch("https://api.kit.forhosting.com/trig/law-of-sines-side", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"side_a": 7,
"angle_a": 40,
"angle_b": 60
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/trig/law-of-sines-side",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"side_a": 7,
"angle_a": 40,
"angle_b": 60
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/trig/law-of-sines-side", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"side_a":7,"angle_a":40,"angle_b":60}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"side_a":7,"angle_a":40,"angle_b":60}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/trig/law-of-sines-side", 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": 7,
"angle_a": 40,
"angle_b": 60
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "trig.law_of_sines_side",
"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 side_a, angle_a, and angle_b, applies b equals a times sin B over sin A in degrees, rejects angle pairs that sum to one hundred eighty or more, and returns the missing side plus the third angle and the common ratio as structured JSON.
What the law of sines says about a missing side from two angles and one side
Returning side_b together with the completed third angle and the explicit ratio a over sin A 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 tables in every language you teach or deploy across homework portals and lightweight engineering scripts that only ever need the missing length.
How this calculator validates input, applies the ratio, and rejects impossible triangles
Angles are measured in degrees as positive interior values strictly less than one hundred eighty. Edge cases are explicit: zero or negative lengths, non-numeric strings, missing fields, angles outside the open interval from zero to one hundred eighty, and angle pairs that cannot close a plane triangle all fail before a fake side is returned. Send a JSON body with three required fields: side_a, angle_a, and angle_b. The engine checks that each field is a finite number, that the known side is positive, and that the sum of the two given angles is strictly less than one hundred eighty so a positive third interior angle exists—if the sum is one hundred eighty or more, no plane triangle exists and the request fails with invalid_input so those failed calls are not charged. The missing side follows from side_b equals side_a times sin of angle_b over sin of angle_a. If either sine is vanishingly small or the resulting length is non-finite or non-positive, the request fails with a clear invalid_input message rather than inventing a fake length. 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 missing-side answer while checking homework, verifying a construction sketch, or confirming a survey note without installing a computer-algebra system. Accepted angles must leave a positive third interior angle, and the known side must be a positive finite length in whatever unit you already use for the figure; the returned missing side shares 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 side_a, angle_a, and angle_b always yield the same side_b, angle_c, and ratio. Wire the endpoint into AAS and ASA drills, contest answer checkers, engineering worksheets that list two angles and one opposite side, or floating preprocessing steps where you already prefer explicit domain errors—especially angle sums that break the triangle—over opaque black-box widgets that cannot show why a labeled figure cannot form a real plane triangle from the given data alone.
Integration use cases
Deterministic fixtures for trigonometry tests
Pipelines that seed unit tests or interactive law-of-sines demos can call a pure missing-side oracle instead of reimplementing degree sine ratios in every language.
Survey and CAD recovery of one opposite length
Field notes that record two interior angles and one opposite side recover the second opposite length as structured JSON for reports or lightweight CAD helpers.
Technical questions
How do I find a missing side with the law of sines using this API?
Send side_a, angle_a (opposite that side), and angle_b to trig.law_of_sines_side. The response returns side_b opposite angle_b, plus angle_c and the common ratio.