Triangle inradius calculator
A 3-4-5 triangle has room for an inscribed circle of radius 1: the calculator divides the area, 6, by half the perimeter, also 6. Enter any three sides and get the radius without angles or heights. 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
- Area
- 6
- Inradius
- 1
The inradius of a triangle is the radius of the unique circle that lies inside the triangle and is tangent to all three sides. When you know only the three side lengths—SSS data from a survey, a textbook exercise, or a CAD edge list—you can recover that radius without measuring angles or altitudes: form the semiperimeter, compute the area with Heron's formula, then divide area by semiperimeter.
What triangle inradius means and how the numbers work
In plane geometry the inradius is the radius of the incircle: the largest circle that fits inside a triangle and touches each side at exactly one point. The incenter—the center of that circle—is also the concurrence of the angle bisectors, but you do not need coordinates or bisectors when only side lengths are known. Label the sides a, b, and c.
First form the semiperimeter s equals (a plus b plus c) divided by two. Then the area A is the positive square root of s times (s minus a) times (s minus b) times (s minus c)—Heron's formula. The inradius is simply r equals A divided by s.
That identity follows from the fact that the triangle splits into three right triangles of heights r whose areas sum to A, so A equals r times s. The product under the Heron radical is zero precisely when the sides are collinear—when one side equals the sum of the other two—so a strict triangle inequality is required: a plus b must exceed c, a plus c must exceed b, and b plus c must exceed a.
A classic sanity check is the three-four-five right triangle: perimeter twelve, semiperimeter six, area six, and inradius one. An equilateral triangle with equal sides six yields semiperimeter nine, area nine times the square root of three, and inradius equal to the square root of three—matching the well-known formula side times root three over six. Every step is elementary arithmetic and one square root on three scalars, so there is no iteration, no sampling, no dependence on wall-clock time, and no need for angles or vertex coordinates in the input contract.
What you can do with it
Game, CAD, and mesh inradius from three edge lengths
Validate a triangle face or compute inscribed-circle proxies when only the three edge lengths are known in the scene graph or mesh data.
FAQ
How do I compute the inradius of a triangle from three sides?
Form semiperimeter s = (a+b+c)/2, area A = sqrt(s(s−a)(s−b)(s−c)), then inradius r = A/s. This tool returns all of those values from three positive side lengths.
What is the triangle inequality?
Each side must be strictly shorter than the sum of the other two: a+b>c, a+c>b, and b+c>a.
Why are collinear sides rejected instead of returning radius zero?
When a+b=c the figure is a line segment, not a triangle.
Does this need angles, height, or vertex coordinates?
No. Only the three side lengths are required. If you have vertices and need the incenter point as well, use the triangle incenter calculator.
Is the triangle inradius calculator free?
The browser tool is free and runs locally.
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-inradius \
-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-inradius", {
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-inradius",
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-inradius", 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-inradius", 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_inradius",
"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 capability checks the strict triangle inequality so the figure is non-degenerate, returns the positive inradius together with area and semiperimeter, and rejects impossible triples with a clear invalid_input error. Results are deterministic JSON—no computer-algebra system, no random seeds, and the same pure solve function in the free browser tool and the prepaid API path—so homework checks, grading scripts, and production pipelines all see the same numbers every time.
What triangle inradius means and how the numbers work
When any of those fails, no triangle exists and this tool returns invalid_input rather than a zero or imaginary radius.
How to call the API and what comes back
Use it from geometry homework pipelines, construction takeoff scripts that need inscribed-circle clearance from three measured edges, game-level tooling, interview-prep sandboxes, or any workflow that needs a reproducible SSS inradius without installing a computer-algebra package on every machine. Prefer this landing when the search intent is specifically the inscribed radius from three sides; if you already have vertex coordinates and need the incenter point as well, use the dedicated triangle incenter calculator instead. Send a JSON body with three fields. Field a is the first positive side length; aliases side_a and side1 are accepted. Field b is the second positive side; aliases side_b and side2 map to the same meaning. Field c is the third positive side; aliases side_c and side3 are accepted. On success the response echoes a, b, and c, adds perimeter, semiperimeter, area, inradius, method set to heron_inradius, and a short formula block documenting the closed forms for s, the Heron radical, and r equals A over s. Failed validation—missing a required field, a non-finite number, a non-positive side, a side outside published limits, or a triple that breaks the strict triangle inequality—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item and zero marginal unit price.
Practical tips, limits, and related searches
When teaching, prefer paired examples: three-four-five for a right triangle with integer inradius one; five-five-six for a simple isosceles check; and one-two-ten to demonstrate the hard rejection path that protects callers from a non-triangle. Prefer this path when you measured or were given all three edges and need the inradius without constructing altitudes. Round-trip checks are easy: after solving, confirm that perimeter equals a plus b plus c, that semiperimeter is half that, that area squared equals s times each (s minus side) within floating tolerance, and that inradius times semiperimeter recovers the area. Keep side lengths inside the published positive range up to one trillion so intermediate products stay finite in IEEE double precision on the Worker and in the browser. Units are whatever you pass—metres, feet, or game units—as long as all three sides share the same unit; the API does not convert units for you. Because the formula only needs sides, students can recompute s, the four Heron factors, the area, and r equals A over s by hand for small integers and match the API line by line. This landing targets people searching for a triangle inradius calculator or an inscribed circle radius from three sides tool; it is not an SAS angle-based solver, not a vertex-based incenter locator, and not a polyhedron insphere tool—use those dedicated capabilities when your data or solid differs. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when users type impossible triples while learning the contract.
Integration use cases
Homework and exam checks for SSS inradius problems
Give students three side lengths; compare their hand-computed semiperimeter, Heron area, and inradius r = A/s against a deterministic API or free browser result.
Surveying and layout clearance from three measured edges
Turn paced or taped edge lengths of a triangular plot into the inscribed-circle radius for clearance, packing, or fixture offsets without measuring a corner angle.