30-60-90 triangle calculator
Solve a 30-60-90 triangle from a single side. A short leg of 5 gives a long leg of 8.660254037844, a hypotenuse of 10 and an area of 21.650635094611. Just say whether you measured the short leg, the long leg or the hypotenuse. Free in your browser.
- Free
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- No sign-up
- Your data stays on your device
Run — free
Worked example
- Side
- 5
- Which
- Short
- Short
- 5
- Long
- 8.6602540378
- Hypotenuse
- 10
- Area
- 21.6506350946
A 30-60-90 triangle is the special right triangle whose angles are fixed at thirty, sixty, and ninety degrees and whose sides always stand in the ratio one to root three to two. Once you know which side you measured—the short leg opposite thirty degrees, the long leg opposite sixty degrees, or the hypotenuse opposite ninety degrees—the other two lengths follow by a single multiply or divide, and the area is half the product of the legs. Students meet this figure in every geometry unit on special right triangles; builders and drafters use it when a right angle meets a sixty-degree miter; game and CAD tools need exact edge lengths without floating-point trigonometry each frame. This capability applies only those fixed ratios: no law of cosines, no iterative solver, no random seeds.
What a 30-60-90 triangle is and how the side ratios work
A thirty-sixty-ninety triangle is a right triangle whose acute angles are fixed: the smaller acute angle is always thirty degrees, the larger is always sixty degrees, and the right angle is ninety degrees. Because the angles never change, the side lengths are locked to a constant proportion that every textbook states as one to the square root of three to two. The shortest side sits opposite the thirty-degree angle; call that length x.
The longer leg, opposite sixty degrees, is then x times the square root of three. The hypotenuse, opposite the right angle, is exactly two x. Area follows at once as one half times short times long, which simplifies to root three over four times x squared when x is the short leg.
Classic check: if the short side is five, the long side is five root three, the hypotenuse is ten, and the area is twenty-five root three over two. If instead the hypotenuse is ten, halve it to recover short equals five and long equals five root three—the same triangle. If the long side is given, divide by root three to recover the short leg, then double for the hypotenuse.
The tool rejects non-positive lengths because a zero or negative side cannot form a triangle; it does not invent a second solution or flip which angle is opposite which side.
Practical tips, limits, and related searches
When teaching, prefer paired examples: short equals one for the unit ratio triangle; short equals five for the classic five, five root three, ten figure; hypotenuse equals ten as a reverse check that recovers the same short and long; and a non-positive side to demonstrate the hard rejection path that protects callers from a degenerate figure. Prefer stating which side you measured before scaling—mixing up short and long is the most common homework error.
Round-trip checks are easy: after solving from short, feed the returned hypotenuse back with which set to hypotenuse and confirm short and long match; likewise feed long with which set to long. Related searches include special right triangles, 45-45-90 companion problems, and ratio of sides in a 30-60-90 triangle; keep those intents on separate landings when the figure class differs.
What you can do with it
Construction and drafting of 30-60-90 miters
Turn a measured short leg, long leg, or diagonal of a thirty-sixty-ninety cut into the other two lengths and the triangular area for takeoff without a separate CAD pass.
Game, CAD, and mesh tooling
Stamp exact 30-60-90 edge lengths into a scene or validate a right-triangle fragment when only one side of the special triangle is already known in the model.
FAQ
What are the side ratios of a 30-60-90 triangle?
The short side is opposite 30°, the long side opposite 60°, and the hypotenuse opposite 90°.
How do I find the other sides if I only know the hypotenuse?
Halve the hypotenuse to get the short side, then multiply the short side by √3 to get the long side. The tool does this when which is set to hypotenuse.
Why must the given side be positive?
A zero or negative length cannot form a triangle.
Does this solve any right triangle, or only 30-60-90?
Only the special 30-60-90 class with fixed ratios. For general SAS data use the SAS triangle calculator; for three free sides use Heron's formula area.
Is the 30-60-90 triangle 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-30-60-90 \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"side":5,"which":"short"}'const res = await fetch("https://api.kit.forhosting.com/geom/triangle-30-60-90", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"side": 5,
"which": "short"
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/triangle-30-60-90",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"side": 5,
"which": "short"
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/triangle-30-60-90", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"side":5,"which":"short"}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"side":5,"which":"short"}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/triangle-30-60-90", 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": 5,
"which": "short"
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.triangle_30_60_90",
"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.
Limits
max_side | 1000000000000 |
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
Results are deterministic JSON from 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 for the same input.
What a 30-60-90 triangle is and how the side ratios work
You never need the law of sines or cosines for a pure 30-60-90 problem once you know which side you were given: scale the unit triangle so that the known side matches, and read off the other two. Every step is elementary scaling on three scalars, so there is no iteration, no sampling, and no dependence on wall-clock time or external libraries beyond ordinary IEEE arithmetic.
How to call the API and what comes back
Use it from geometry homework pipelines, construction takeoff scripts that already know one edge of a 30-60-90 miter, game-level tooling that stamps special right triangles into a mesh, interview-prep sandboxes, or any workflow that needs a reproducible special-right-triangle solve without installing a computer-algebra package on every machine. Prefer this landing when the figure is known to be 30-60-90; for a general two-sides-and-included-angle problem use the SAS sibling, and for three free sides use Heron area. Send a JSON body with two fields. Field side is the positive length of the known edge; aliases length, value, and s are accepted. Field which names the role of that edge and must be one of short, long, or hypotenuse; aliases of, side_type, given, known, hyp, opposite_30, opposite_60, and opposite_90 map to the same three roles. On success the response returns short, long, and hypotenuse after fixed twelve-decimal rounding, area, perimeter, a fixed angles object documenting thirty opposite short, sixty opposite long, and ninety opposite the hypotenuse, a ratios block, given and given_side echoing the request, method set to 30-60-90, and a short formula block for the closed forms. Failed validation—missing a required field, a non-finite number, a non-positive side, or an unrecognized which label—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
Keep the known side 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. Because long is short times root three and hypotenuse is twice short, students can recompute by hand for small integers and match the API line by line within floating tolerance after the fixed rounding. This landing targets people searching for a 30-60-90 triangle calculator, a special right triangle sides solver, or how to find the missing sides of a thirty-sixty-ninety triangle from one given length; it is not a general triangle solver and will not accept free angle inputs. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when users type zero or negative lengths while learning the contract.
Integration use cases
Homework and exam checks for special right triangles
Give students one side of a 30-60-90 triangle and which role it plays; compare their hand-computed missing sides and area against a deterministic API or free browser result.