Equilateral triangle area from side calculator
Enter one side and get the equilateral triangle's area, height and perimeter at once: side 6 gives an area of 9√3, about 15.5885, a height of about 5.1961 and a perimeter of 18. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Side
- 6
- Area
- 15.5884572681
- Height
- 5.1961524227
- Perimeter
- 18
Enter the side length of an equilateral triangle and get its area instantly from the closed-form formula A = (√3/4)·s², plus height and perimeter for layout checks. The side must be a positive finite number in any consistent length unit; the area is returned in the corresponding square units.
What this calculator returns from a single side
An equilateral triangle has three equal sides and three equal sixty-degree angles, so one positive length fully determines the shape up to placement in the plane. This capability takes that single side length and returns the geometric area together with the altitude and the perimeter, so you can verify the result against sketch paper or CAD without running a second tool. The primary output is area, computed with the exact closed form A equals the square root of three over four, times the side squared.
Height follows as the square root of three over two times the side, and perimeter is simply three times the side. All three numbers share the same input units: if the side is in metres, area is square metres and height remains in metres.
The equilateral area formula and why it is exact
Drop an altitude from any vertex of an equilateral triangle to the opposite side. That altitude bisects the base into two segments of length s over two and creates a right triangle with hypotenuse s and base s over two. Pythagoras then gives height h equals the square root of s squared minus s squared over four, which simplifies to s times the square root of three over two.
The area of any triangle is one half base times height, so A equals one half times s times h, which expands to one half times s times s times the square root of three over two, or equivalently the square root of three over four times s squared. There is no iterative root finding, no random sampling of the plane, and no dependence on the current time or locale.
For a worked check, side six yields area nine times the square root of three, about fifteen point five eight eight five, height three times the square root of three, about five point one nine six one, and perimeter eighteen. Side two yields area equal to the square root of three, matching the familiar textbook case where area and height share the same numeric value only because s equals two.
When you scale the side by a factor k, area scales by k squared and height and perimeter scale by k, which is exactly the dimensional behaviour you expect from a planar region.
What to enter and which units you get back
Validation runs before any formula evaluation. Values less than or equal to zero are rejected: a negative length is not a geometric side, and a zero length collapses the triangle to a point with zero area that is better treated as bad input than as a silent success. Extremely small sides below one times ten to the minus fifteen and sides above one times ten to the twelve are rejected to keep results inside a stable double-precision window and to block accidental unit mistakes such as nanometres mixed with light-years.
Units are caller-defined and never converted: send centimetres and you receive square centimetres; convert yourself if you need square feet. Failed validation is not charged.
What you can do with it
Classroom geometry checks
Students enter one side length and compare the computed area and height against hand work using A = (√3/4)·s² without needing a full triangle solver.
Fabrication and laser-cut equilateral panels
Shop drawings often specify only the edge length; convert that length to panel area for material estimates and nesting before cutting.
Game and graphics layout
Procedural maps and UI badges that place equilateral tiles need area and height from side length for bounding boxes and density budgets.
FAQ
What input do I provide?
Units are yours; area is returned in those units squared.
What formula is used for the area?
A = (√3/4)·s². Height is (√3/2)·s and perimeter is 3·s. All three are derived from the same side in one deterministic pass.
What happens if the side is negative?
A negative length is not a valid equilateral side, so no area is returned and the task is not charged.
Can I send zero or omit the side?
The triangle must have a strictly positive side length.
How do I find the height of an equilateral triangle?
Multiply the side by the square root of three and divide by two. For side 6 that is 3√3, about 5.1961, and it appears next to the area so you can check a sketch at a glance.
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/equilateral-area \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"side":6}'const res = await fetch("https://api.kit.forhosting.com/geom/equilateral-area", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"side": 6
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/equilateral-area",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"side": 6
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/equilateral-area", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"side":6}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"side":6}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/equilateral-area", 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": 6
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.equilateral_area",
"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
Negative or zero sides are rejected with a clear invalid_input error so bad data never becomes a silent zero.
What this calculator returns from a single side
Optional aliases s, a, and length are accepted for the same field so existing form payloads map without renaming keys. Missing, non-numeric, infinite, zero, or negative values fail fast with invalid_input and are never billed.
The equilateral area formula and why it is exact
That identity is algebraic, not approximate: the only floating-point step is evaluating the square root of three once in IEEE double precision and multiplying.
Validation, units, limits, and how to use the API
The body must supply side, or one of the aliases s, a, or length, as a finite number or a numeric string. Typical integrations post a tiny JSON object with a single side field, read the area field from the response, and optionally log height and perimeter for audit trails in manufacturing or classroom gradebooks.
Technical questions
How much does the API cost?
Successful calls are priced at $0.002 per item. The browser widget is free for interactive use; failed validation is not billed.