30-60-90 triangle sides
In a 30-60-90 triangle with a short side of 5, the long side is 5√3 and the hypotenuse is 10. Enter one side, say whether it is the short leg, the long leg or the hypotenuse, and get the other two. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Side
- 5
- Position
- Short
- Short
- 5
- Long
- 8.6602540378
- Hypotenuse
- 10
- Given side
- 5
A 30-60-90 triangle is a right triangle whose acute angles are thirty degrees and sixty degrees, so the sides stand in one fixed ratio for every similar copy: the short leg opposite thirty degrees, the long leg opposite sixty degrees, and the hypotenuse opposite ninety degrees always compare as one to the square root of three to two. Homework, drafting, and exam items almost always give exactly one of those three lengths and name the position it occupies, then ask for the other two. This 30-60-90 triangle side calculator accepts that single positive finite length and its position — short, long, or hypotenuse — applies the closed-form ratios, and returns all three side lengths.
What a 30-60-90 triangle is and how the three sides appear
A 30-60-90 triangle is the special right triangle obtained by cutting an equilateral triangle in half along an altitude. The cut creates a right angle, leaves a sixty-degree angle at the uncut vertex, and splits the opposite sixty-degree angle into two thirty-degree angles. Because the original equilateral sides were equal, the new hypotenuse is twice the short leg that used to be half a base, and the altitude that became the long leg is the short leg times the square root of three.
Those three facts collapse into a single side ratio that never changes under similarity: short to long to hypotenuse equals one to the square root of three to two. The short side always sits opposite the thirty-degree angle, the long side opposite the sixty-degree angle, and the hypotenuse opposite the right angle.
Given any one of those three lengths and the position it occupies, the other two sides are a single multiply or divide: if the short side is known, the long side is that length times the square root of three and the hypotenuse is twice the short side; if the long side is known, the short side is that length divided by the square root of three and the hypotenuse is twice the short side; if the hypotenuse is known, the short side is half of it and the long side is that half times the square root of three.
Classic classroom values appear immediately: a short side of five yields a long side of five times the square root of three and a hypotenuse of ten. A hypotenuse of ten yields a short side of five and the same long side. Because a Euclidean length cannot be zero or negative, the given side must be a positive finite number; a signed cut, Infinity, or a non-numeric token is not a geometric side and must not look like a successful three-side solve.
This calculator encodes that contract and returns the three lengths together with the given position and the ratio identity.
Where 30-60-90 side lengths come in handy
Classrooms and tutoring platforms use a 30-60-90 triangle sides calculator so learners can verify the two unknown lengths without mixing up which ratio to write first. Exam generators and LMS plugins need a deterministic oracle that rejects a non-positive given side instead of taking absolute values and marking a signed homework token as correct. Construction and layout tools convert one measured member of a 30-60-90 cut — a short run, a long rise, or a hypotenuse brace — into the other two stock lengths before ordering material.
Roof, ramp, and stair sketches often freeze a thirty-sixty-ninety profile: one known length and its position fix the remaining two without reopening a trigonometry table. Game engines and physics sandboxes spawn a similar copy from whichever one side the designer already stored. Keep requests small, pin the rounded short, long, and hypotenuse in tests, and you get reproducible geometry for homework, edge workers, and design scripts without standing up symbolic math infrastructure of your own.
The landing is deliberately narrower than a general right-triangle solver: it does not invent an angle, it does not apply Pythagoras to two arbitrary legs, and it does not return area as the primary product, so the page stays aligned with the search for the three side lengths of a 30-60-90 triangle from one given side and its position.
What you can do with it
Classroom 30-60-90 checks
Students enter one known side and whether it is short, long, or the hypotenuse to verify all three lengths against textbook 1 : √3 : 2 exercises.
Exam and LMS auto-grading
Mark 30-60-90 missing-side items with a deterministic oracle that rejects a non-positive given side and never invents extra lengths.
One measured cut to the other two
Convert a measured short run, long rise, or hypotenuse brace on a 30-60-90 profile into the remaining two stock lengths before ordering material.
FAQ
What are the side ratios of a 30-60-90 triangle?
The short side is opposite 30°, the long side is opposite 60°, and the hypotenuse is opposite 90°.
What happens if the given side is zero or negative?
A Euclidean side must be a positive finite number; zero, negatives, Infinity, and non-numeric tokens are not billed.
Does this calculator return area or angles?
No. It returns the three side lengths, the echoed given side and position, and the ratio formula. Area and a general right-triangle solve are separate landings.
How do I find the other sides when I know the hypotenuse?
Halve the hypotenuse to get the short side, then multiply that half by √3 for the long side. A hypotenuse of 10 gives a short side of 5 and a long side of 5√3.
Why is the hypotenuse twice the short side?
Because a 30-60-90 triangle is half of an equilateral triangle cut along its height. The hypotenuse is a full side of the original triangle, while the short side is half of its base.
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/triangle-30-60-90 \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"side":5,"position":"short"}'const res = await fetch("https://api.kit.forhosting.com/math/triangle-30-60-90", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"side": 5,
"position": "short"
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/triangle-30-60-90",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"side": 5,
"position": "short"
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/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,"position":"short"}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"side":5,"position":"short"}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/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,
"position": "short"
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.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.
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
Zero, negative, infinite, and non-numeric sides fail with invalid_input and are not billed.
Input contract, response shape, and error behaviour
Example: side equals five and position equals short yields short five, long five times the square root of three, and hypotenuse ten. Example: side equals ten and position equals hypotenuse yields the same three lengths from the opposite end of the ratio. POST a JSON body with two required fields. The field side is the known length and must be a positive finite number; it also accepts the aliases length, value, and given_side, and it may arrive as a JSON number or as a numeric string. The field position names which role that length already occupies and must be one of the closed set short, long, or hypotenuse; it also accepts the aliases which, of, and given, plus the classroom tokens opposite_30, opposite_60, opposite_90, and hyp. Missing either field, sending an empty string, or sending a position outside that set fails with invalid_input naming what was expected. A given side that is zero, negative, infinite, NaN, or not a decimal number fails with invalid_input stating that a positive finite number is required. Those validation failures are not billed. On success the body returns short, long, and hypotenuse as the three side lengths rounded to twelve decimal places for stable goldens across runtimes, echoes given_side and position, and restates the identity in the formula field as short colon long colon hypotenuse equals one colon the square root of three colon two. Pair the structured JSON with your own drawing code if you need a sketch; this endpoint returns the three side lengths only and does not emit SVG, angles as a separate product, area, or unit conversion — you supply the given length already in one consistent unit.
Technical questions
How do I find the sides of a 30-60-90 triangle from one length?
Send a positive finite side and its position: short (opposite 30°), long (opposite 60°), or hypotenuse (opposite 90°). The API returns all three lengths from the fixed 1 : √3 : 2 ratios.
What does the 30-60-90 triangle sides calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.