Triangle vertices calculator
The points (0, 0), (3, 0) and (0, 4) form a triangle with sides of 3, 4 and 5, a perimeter of 12 and an area of 6. Enter the coordinates of three vertices on a plane and get all of these at once. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- X1
- 0
- Y1
- 0
- X2
- 3
- Y2
- 0
- X3
- 0
- Y3
- 4
- Area
- 6
- Perimeter
- 12
- Side ab
- 3
- Side bc
- 5
A triangle on the Cartesian plane is completely determined by the coordinates of its three vertices, and those six numbers already contain the area, the three side lengths, and the closed perimeter. This calculator accepts the planar coordinates of vertices A, B, and C, rejects any coordinate that is missing, infinite, or not a number, and rejects collinear triples that do not enclose a positive area. On a valid triple it returns the shoelace area, the Euclidean lengths of sides AB, BC, and CA, and the perimeter as their sum.
How area, sides, and perimeter come from three vertices
Start with three ordered points A = (x1, y1), B = (x2, y2), and C = (x3, y3). The signed quantity twice the area is the shoelace sum x1 times (y2 minus y3) plus x2 times (y3 minus y1) plus x3 times (y1 minus y2). The geometric area of the triangle is one half of the absolute value of that sum.
Independently of winding, each side is the Euclidean distance between a pair of vertices. Side AB is the hypot of (x2 minus x1) and (y2 minus y1); side BC uses the same hypot on B and C; side CA closes the figure from C back to A. Because hypot is even in each difference, listing the vertices clockwise or counterclockwise never changes a length.
The perimeter is the sum of those three positive lengths. Together the area, the three sides, and the perimeter are the complete first-order measures of a planar triangle given only by vertices: you do not need a named side, an included angle, or a height. A classic classroom check is the right triangle with vertices at (0, 0), (3, 0), and (0, 4): the sides are 3, 4, and 5, the perimeter is 12, and the area is 6.
An isosceles check is (0, 0), (2, 0), (1, 1): the base is 2, the equal legs are each the square root of 2, the perimeter is 2 plus twice the square root of 2, and the area is 1. The path is constant time, uses no trigonometric library, and is fully deterministic on the six numbers you provide.
What you send and what you get back
Provide six required finite numbers: x1 and y1 for vertex A, x2 and y2 for vertex B, and x3 and y3 for vertex C. After parsing, the service evaluates the shoelace sum and the three hypot lengths. If the three points are collinear the error message states that the vertices are collinear and asks for a non-degenerate triple.
If an intermediate overflows to a non-finite value the message asks you to reduce coordinate magnitudes. Example checks: zero, zero, three, zero, zero, four yields area six, perimeter twelve, and sides three, five, and four; zero, zero, one, one, two, two fails as collinear; a missing y3 fails naming that field. Use the free browser tool for a quick sanity check, then call the identical contract from scripts when you need bulk evaluation of coordinate-based triangles without reimplementing the collinear gate in every language.
Where vertex measures show up in real work
Analytic geometry courses ask students to recover every elementary measure of a plotted triangle from the three marked points, not just one of them. A single calculator that returns area, sides, and perimeter together, and that refuses a collinear triple, cuts the usual spreadsheet mistakes: swapping a coordinate, forgetting the half in the shoelace formula, adding only two hypotenuses, or treating a flattened figure as a valid answer.
Local-plane drafting and parcel work already store corners as projected (x, y) pairs; the same six numbers decide how much paint, fence, or trim a three-corner lot needs and how much ground it covers, so field notes and office scripts should share one numeric oracle instead of three ad-hoc formulas. Game and CAD tutorials often keep mesh faces as triples of screen or world coordinates; packing area, edge lengths, and perimeter into one response feeds collision proxies, UV density checks, and level-of-detail heuristics without standing up a computational-geometry library.
Dedicated landings already exist when the search is only for area from coordinates, only for perimeter from vertices, spherical length on the globe, or a semiperimeter from three already-known sides; those are different queries, and this page stays the combined triangle-vertices calculator.
What you can do with it
Homework and exam checks
Recover area, three sides, and perimeter from plotted vertices, and reject collinear triples instead of accepting a silent degenerate zero.
Local-plane survey parcels
From three projected corner coordinates, recover planar area and edge measures before reporting, staking, or ordering materials for a triangular lot.
FAQ
What happens if the three vertices are collinear?
Collinear points do not form a triangle, so a zero-area or degenerate walk is not returned as a successful solve.
Does the order of the three vertices change the result?
Area is taken as an absolute value and each distance is symmetric, so listing the vertices clockwise or counterclockwise yields the same area, the same three lengths, and the same perimeter.
Are the coordinates two-dimensional only?
Yes. This tool expects planar Cartesian x and y values. It does not compute 3D space-triangle measures or geographic great-circle length.
What does the triangle vertices calculator cost?
The browser tool is free and runs locally.
How is the area worked out from the coordinates?
With the shoelace formula: multiply each x by the difference of the other two y values, add the three products and take half of the absolute value. Each side is the straight-line distance between two corners.
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-vertices \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"x1":0,"y1":0,"x2":3,"y2":0,"x3":0,"y3":4}'const res = await fetch("https://api.kit.forhosting.com/math/triangle-vertices", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"x1": 0,
"y1": 0,
"x2": 3,
"y2": 0,
"x3": 0,
"y3": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/triangle-vertices",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"x1": 0,
"y1": 0,
"x2": 3,
"y2": 0,
"x3": 0,
"y3": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/triangle-vertices", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"x1":0,"y1":0,"x2":3,"y2":0,"x3":0,"y3":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"x1":0,"y1":0,"x2":3,"y2":0,"x3":0,"y3":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/triangle-vertices", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"x1": 0,
"y1": 0,
"x2": 3,
"y2": 0,
"x3": 0,
"y3": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.triangle_vertices",
"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
There are no network calls, no random samples, and no clocks: only IEEE-754 arithmetic on the six coordinates you send.
How area, sides, and perimeter come from three vertices
When the signed sum is exactly zero the three points lie on a single straight line: they do not form a triangle, and this calculator refuses the request instead of returning a successful area of zero that would look like a solved figure. Very large coordinates can still overflow the IEEE-754 double range; when an intermediate sum, a length, or the area is non-finite the request fails with a clear overflow message rather than serializing Infinity as JSON.
What you send and what you get back
Each field may arrive as a JSON number or as a numeric string with an optional sign, decimal point, and exponent; empty strings, bare signs, non-numeric text, NaN, and Infinity fail with invalid_input naming the field and are not charged. On success the JSON response returns area and perimeter rounded for stable serialization, the three side lengths side_ab, side_bc, and side_ca, the six normalized coordinates, and method set to the string vertices_measures so logs and teaching interfaces can document the algorithm. API pricing is the published base of $0.002 per successful item.
Where vertex measures show up in real work
Because this capability is a pure parse engine, failed validation does not consume balance and successful runs stay at $0.002 per item.
Integration use cases
Game and CAD face helpers
Turn three 2D vertex pairs into area, sides, and perimeter for mesh diagnostics, using the same vertices_measures path in browser and API.
Technical questions
What does the triangle vertices calculator return?
It returns the shoelace area, the three Euclidean side lengths side_ab, side_bc, and side_ca, the closed perimeter, the six normalized coordinates, and method set to "vertices_measures".