Polygon area calculator
Paste the corners of a polygon in order and get its area: the rectangle (0, 0), (4, 0), (4, 3), (0, 3) gives 12, and the sign of the area tells you whether you listed them clockwise or counterclockwise. Free in your browser.
- Free
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- Your data stays on your device
Run — free
Runs in your browser. Free, unlimited — your data never leaves this page.
This calculator turns an ordered list of planar coordinates into the area of a simple polygon with the classic shoelace—or surveyor's—formula used in surveying, GIS prep work, CAD checks, and programming contests. You paste vertices as [x, y] pairs or {x, y} objects in boundary order; the engine closes the ring for you if you omit the repeated first point, drops a trailing duplicate when you already closed it, and returns the absolute area plus the signed area that reveals winding direction.
How the shoelace sum is evaluated step by step
Each entry is parsed to finite x and y; decimal strings are allowed for form callers. If the last point equals the first, that closing duplicate is dropped so callers who paste a closed ring and callers who omit the repeat share one code path. Signed area is half of that sum; absolute area is the magnitude.
Orientation is counterclockwise when the signed area is positive, clockwise when negative, and degenerate when the signed area is exactly zero (collinear or zero-area rings). Optional precision only rounds the published fields; the internal solve always uses full double precision before that final rounding step.
Where polygon area from vertices fits in study, design, and code
Students meet the shoelace formula when a textbook asks for the area of a quadrilateral given corner coordinates, when an exam demands the absolute area of a clockwise triangle, or when analytic geometry homework wants both signed and absolute area before discussing orientation. Teachers can pre-generate answer keys that list area for integer vertex sets and that refuse rings with fewer than three points instead of printing zero.
Surveying and GIS prep often need a quick check of a parcel polygon before projection-aware software is involved; knowing that a pasted ring has only two distinct corners is as valuable as the number itself. Floor-plan and CAD workflows extract vertex chains from walls or room outlines and need a deterministic square footage check for scripts and code reviews. Game and simulation teams use ordered polygon areas for collision hulls, region triggers, and level design tools that should stay identical across platforms.
What you can do with it
Homework and exam preparation
Compute polygon area from textbook vertex lists with the shoelace formula, check winding via signed area, and verify that fewer than three points are rejected.
CAD, floor plans, and GIS prep
Quick-check room or parcel outlines from ordered corners before heavier projection-aware tooling, with honest errors on incomplete rings.
FAQ
What does the polygon area calculator cost?
It is free in your browser on this page.
Do I need to repeat the first vertex at the end?
No. The ring is closed automatically. If you do repeat the first point, that trailing duplicate is dropped before counting vertices.
Does vertex order matter?
Absolute area is the same for either winding on a simple polygon.
What does a negative signed area mean?
It means you listed the corners clockwise. A positive value means counterclockwise, and exactly zero points to collinear corners or a shape with no area. The absolute area is the same in both directions.
Why do I get an error with only two points?
A polygon needs at least three distinct corners. If fewer than three remain after the repeated closing point is dropped, the calculator reports an error instead of returning a misleading zero.
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/polygon-area-shoelace \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"vertices":[[0,0],[4,0],[4,3],[0,3]]}'const res = await fetch("https://api.kit.forhosting.com/geom/polygon-area-shoelace", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"vertices": [
[
0,
0
],
[
4,
0
],
[
4,
3
],
[
0,
3
]
]
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/polygon-area-shoelace",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"vertices": [
[
0,
0
],
[
4,
0
],
[
4,
3
],
[
0,
3
]
]
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/polygon-area-shoelace", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"vertices":[[0,0],[4,0],[4,3],[0,3]]}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"vertices":[[0,0],[4,0],[4,3],[0,3]]}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/polygon-area-shoelace", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"vertices": [
[
0,
0
],
[
4,
0
],
[
4,
3
],
[
0,
3
]
]
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.polygon_area_shoelace",
"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
When fewer than three vertices remain after that normalization, or any coordinate is missing or non-finite, the tool returns invalid_input instead of a silent zero or NaN. Through the API a successful request costs $0.002; validation failures are not charged.
What a shoelace polygon-area calculator returns and why order matters
A simple polygon is a closed chain of straight edges that does not cross itself. Its area can be recovered from the sequence of vertex coordinates alone: you do not need triangle decomposition, raster fill, or Monte Carlo sampling when the boundary is already ordered. The shoelace formula multiplies each consecutive pair of coordinates in a criss-cross pattern, sums the signed contributions, halves the total, and takes the absolute value so clockwise and counterclockwise rings report the same positive area. If you need ellipse metrics from semi-axes, a circle through three points, or triangle solvers, use a tool dedicated to those problems; the search intent here is polygon area from ordered vertices via shoelace, with a clear answer or an honest error when the ring is too short. This tool exists to apply that identity deterministically from a JSON vertex list without forcing you to open a spreadsheet or re-implement ring closure. Preferred input is the array field vertices; aliases points, polygon, and coords are accepted for form and spreadsheet callers. Each element may be a two-number array [x, y] or an object with x and y (lon/lat style keys are also recognized for GIS-shaped payloads). The response always includes n (vertex count after normalization), area, signed_area, orientation (counterclockwise, clockwise, or degenerate), the formula string, and the normalized vertices with rounded coordinates. Missing vertices, a non-array payload, non-finite coordinates, or fewer than three points after dropping a closing duplicate all raise invalid_input with a message that names the failure. That discipline keeps automated tests stable, makes caching safe, and prevents a script from treating a line segment as a filled region.
How the shoelace sum is evaluated step by step
The engine resolves the vertex list from vertices with aliases points, polygon, and coords, requiring a real array. After that step the list must still contain at least three points; two or fewer points raise invalid_input because a polygon area is undefined for a segment or a lone point. The shoelace sum walks i from zero to n minus one, using modular arithmetic so the successor of the last vertex is the first: each term is x_i times y of the next minus x of the next times y_i. Pair the free browser run on this page with the priced API when you need the identical mathematical result inside an automated pipeline without re-implementing ring normalization or the surveyor's sum yourself.
Where polygon area from vertices fits in study, design, and code
On this page the work runs entirely in your browser; through the API each successful lookup costs $0.002, failed validation is not charged, and nothing is stored after the response returns. That split—free interactive exploration and a priced, identical API—matches the rest of the KIT tier-A catalog and lets you move from a classroom demo to an automated pipeline without changing the mathematical result. Pair it with other free geometry endpoints when you need ellipse metrics, circle construction, or triangle solvers for a lab of deterministic tools, and with documentation that shows both the vertex list and the finished area so the numbers and the picture stay aligned for learners and reviewers. Whether you arrived searching for a polygon area calculator or for the shoelace formula online, the contract is the same: ordered vertices in, finite area or a clear invalid_input out.
Integration use cases
Deterministic geometry fixtures
Generate reproducible polygon areas for unit tests, contest validators, and CI assertions without hand-rolled ring closure or surveyor's sums.
Technical questions
What if I provide fewer than three vertices?
The handler returns invalid_input with a message that at least three vertices are required for a polygon area.
Can I control rounding?
Yes. Set precision (aliases decimals, dp) to an integer from 0 to 15; the default is 10 decimal places on all numeric fields.