Triangle angle bisector lengths
Get the three internal angle bisector lengths from the sides alone: a 3-4-5 triangle has bisectors of 4.216370213558, 3.35410196625 and 2.424366106925. No angles or coordinates needed, and impossible side sets are refused. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- A
- 3
- B
- 4
- C
- 5
- Perimeter
- 12
- Semiperimeter
- 6
- Bisector a
- 4.2163702136
- Bisector b
- 3.3541019663
The internal angle bisectors of a triangle are the three rays that split each vertex angle in half and meet at the incenter. Their lengths—from each vertex to the opposite side—are fully determined by the three side lengths alone, with no need for angles or coordinates.
What angle-bisector lengths mean and the side-only formula
In Euclidean plane geometry every non-degenerate triangle has three internal angle bisectors. Each bisector starts at a vertex, cuts that angle into two equal parts, and meets the opposite side at a point given by the angle-bisector theorem: it divides the opposite side in the ratio of the adjacent sides. The length of that segment—from vertex to opposite side—is a scalar students and engineers often need without constructing the figure.
Search intent clusters around phrases such as triangle angle bisector lengths calculator, length of angle bisector from three sides, SSS angle bisector formula, and find internal bisector lengths a b c. Learners want numbers they can check against a homework key before drawing; pipeline tools want the same numbers as stable intermediates for incenter construction, clearance, or mesh diagnostics. Units are whatever you pass—metres, feet, or game units—as long as all three sides share the same unit; the engine does not convert units.
Input contract, validation, and how the engine applies the formula
Each must be a finite positive number; numeric strings that parse cleanly are accepted so form posts and spreadsheet exports stay convenient. The engine then enforces the strict triangle inequality: a plus b greater than c, a plus c greater than b, and b plus c greater than a. Degenerate or impossible side triples fail with a clear message about the triangle inequality rather than returning a silent zero or a complex intermediate under the square root.
Missing fields, empty strings, Infinity, and NaN also fail before any arithmetic runs. There is no unit conversion inside the capacity: if you send centimetres you get centimetres of bisector length. Failed validation is not billed; only a successful item consumes the published base price.
What you can do with it
SSS contest problems and construction takeoffs
Normalize three measured or given edges into internal bisector lengths for layout checks, clearance estimates, and textbook identity verification without needing angles.
LMS and contest auto-grading for bisector lengths
Mark angle-bisector items with a pure oracle that rejects triangle-inequality failures and never invents floating noise beyond fixed decimal rounding.
FAQ
What happens if the sides violate the triangle inequality?
Degenerate or impossible triples have no well-defined positive internal bisector lengths, so the tool refuses to invent values.
Do I need angles or vertex coordinates?
No. Only the three side lengths a, b, and c are required. The closed-form formulas depend only on the sides.
Is the triangle angle bisector lengths calculator free?
The browser tool is free and runs locally.
How do I find the length of an angle bisector from three sides?
Multiply the two sides that meet at the vertex by one minus the square of the opposite side over their sum, then take the square root. For a 3-4-5 triangle the bisectors are 4.216370213558, 3.35410196625 and 2.424366106925.
Where does an angle bisector meet the opposite side?
At the point that splits the opposite side in the same ratio as the two sides next to the angle, by the angle bisector theorem. The three bisectors also cross at one point, the incenter.
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-angle-bisectors \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"a":3,"b":4,"c":5}'const res = await fetch("https://api.kit.forhosting.com/geom/triangle-angle-bisectors", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"a": 3,
"b": 4,
"c": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/triangle-angle-bisectors",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"a": 3,
"b": 4,
"c": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/triangle-angle-bisectors", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"a":3,"b":4,"c":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"a":3,"b":4,"c":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/triangle-angle-bisectors", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"a": 3,
"b": 4,
"c": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.triangle_angle_bisectors",
"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 triangle angle bisector lengths calculator applies the classical closed form t_a equals the square root of b times c times one minus the square of a over b plus c, and the two cyclic companions for t_b and t_c. You send positive sides a, b, and c that satisfy the strict triangle inequality; the engine returns the three bisector lengths as structured JSON together with the perimeter, semiperimeter, echoed sides, and a short formula string. The same pure function powers the free browser tool and the prepaid API path, so homework keys, grading scripts, CAD preprocessors, and edge workers never disagree on a three-side SSS bisector triple, and no language model or external geometry service sits in the computation path.
What angle-bisector lengths mean and the side-only formula
With sides a, b, and c opposite vertices A, B, and C respectively, the length of the bisector from A is t_a equals the square root of b times c times the quantity one minus a over b plus c, all squared. The formulas for t_b and t_c follow by cycling the letters. Equivalent forms use the semiperimeter, such as t_a equals two times the square root of b c s times s minus a, all divided by b plus c; both agree for any valid triangle. Delivering a, b, c, perimeter, semiperimeter, bisector_a, bisector_b, bisector_c, and a formula field as JSON makes the result easy to paste into notes, pin in automated tests, or feed into a larger geometry pipeline without reimplementing the radicand in every language.
Input contract, validation, and how the engine applies the formula
POST a JSON body with three required fields: a, b, and c for the three side lengths. Aliases side_a, side1, side_b, side2, side_c, and side3 map to the same three inputs for classroom templates that already label sides that way. Zero or negative values fail immediately with invalid_input naming the field, because a Euclidean side length must be strictly greater than zero.
Practical uses, pricing, limits, and when to call related tools
Use the free browser tool when you want immediate internal angle-bisector lengths while checking homework, preparing a construction layout, verifying an SSS contest problem, or confirming three measured edges before a takeoff. Limits are intentional: only three planar side lengths, no angles, no coordinates, no external bisectors, and no unit conversion tables. If you need the incenter coordinates, send vertices to an incenter tool; if you need only the semiperimeter or area, choose those dedicated capabilities instead. Each successful request is one billable item at $0.002; invalid input does not consume balance. Pair this endpoint with a dedicated incenter calculator when you also need the meeting point of the bisectors from coordinates, or with Heron area when you need area from the same three sides: those landings answer different search intents—bisector lengths from sides versus incenter from vertices versus SSS area—so keeping them focused improves SEO clarity and keeps each response small.
Integration use cases
Homework and exam angle-bisector length drills
Give students three side lengths; compare their hand-computed t_a, t_b, and t_c against a deterministic API or free browser result before they construct the figure.
Technical questions
How do I find the angle bisector lengths of a triangle from three sides?
Enter positive sides a, b, and c. The calculator returns bisector_a, bisector_b, and bisector_c using t_a = sqrt(b*c*(1-(a/(b+c))^2)) and the cyclic formulas for t_b and t_c.
What is t_a versus t_b and t_c?
bisector_a (t_a) is the length of the internal bisector from the vertex opposite side a (angle A). bisector_b and bisector_c are the bisectors from the other two vertices.