Quadrilateral area calculator
Type the four corners of a quadrilateral in boundary order and get the area it encloses: (0, 0), (4, 0), (4, 3), (0, 3) give 12, with a perimeter of 14 and sides of 4 and 3. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- X-coordinate of vertex 1
- 0
- Y-coordinate of vertex 1
- 0
- X-coordinate of vertex 2
- 4
- Y-coordinate of vertex 2
- 0
- X-coordinate of vertex 3
- 4
- Y-coordinate of vertex 3
- 3
- Area
- 12
- Signed area
- 12
- Perimeter
- 14
- Side lengths
- 4, 3, 4, 3
A quadrilateral area calculator turns four corner coordinates into a single positive area without measuring diagonals or heights by hand. Surveyors, CAD and game pipelines, GIS scripts, and geometry students often already have the vertices of a plot, room, or polygon face and only need the enclosed area. This capability applies the classic shoelace formula to four points listed in sequential boundary order—clockwise or counterclockwise—checks that the figure is a simple non-self-intersecting quadrilateral, and returns area, signed area, side lengths, perimeter, and orientation.
What the shoelace formula is and why vertex order matters
The shoelace formula, also called the surveyor's formula, computes the area of a simple polygon from an ordered list of vertices in the plane. For a quadrilateral with vertices (x1,y1), (x2,y2), (x3,y3), and (x4,y4), form the alternating sum x1 y2 minus x2 y1, plus x2 y3 minus x3 y2, plus x3 y4 minus x4 y3, plus x4 y1 minus x1 y4; half the absolute value of that sum is the area.
The sign of the sum encodes orientation: positive for a counterclockwise walk around the boundary and negative for clockwise, which this tool reports as orientation while always returning a non-negative area. Vertices must be listed in sequential order around the edge—each point followed by its true boundary neighbour—not as two diagonals or a shuffled corner list. A classic sanity check is the axis-aligned unit square with corners (0,0), (1,0), (1,1), and (0,1): perimeter four and area one.
A three-by-four rectangle with corners (0,0), (4,0), (4,3), and (0,3) yields area twelve, matching base times height. Every step is elementary arithmetic on eight scalars, so there is no iteration, no sampling, no wall-clock dependence, and no need for side lengths or angles in the input contract when coordinates are already known.
What you can do with it
Surveying and room takeoff from measured corners
Turn taped or GPS corner points of a four-sided plot or room into floor or land area without measuring diagonals or constructing altitudes.
Game, CAD, and mesh face area from four vertices
Validate a quad face or compute surface area when four corner positions are known in the scene graph, mesh, or level editor.
FAQ
What is the shoelace formula for a quadrilateral?
List four vertices in order. Area is half the absolute value of Σ (xi yi+1 − xi+1 yi), closing the loop so the fifth point equals the first. The sign encodes clockwise versus counterclockwise orientation.
Do vertices have to be in a specific order?
Yes. List them sequentially around the boundary (clockwise or counterclockwise). Shuffled or diagonal-first orders can self-intersect or measure the wrong region and are rejected when opposite edges cross.
What happens with a self-intersecting bow-tie quadrilateral?
If opposite edges cross, the figure is not a simple quadrilateral.
Does this need side lengths or angles?
No. Only the eight coordinates of the four vertices are required. If you have three side lengths of a triangle instead, use a Heron area calculator.
Is the quadrilateral area 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/quadrilateral-area \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"x1":0,"y1":0,"x2":4,"y2":0,"x3":4,"y3":3,"x4":0,"y4":3}'const res = await fetch("https://api.kit.forhosting.com/geom/quadrilateral-area", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"x1": 0,
"y1": 0,
"x2": 4,
"y2": 0,
"x3": 4,
"y3": 3,
"x4": 0,
"y4": 3
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/quadrilateral-area",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"x1": 0,
"y1": 0,
"x2": 4,
"y2": 0,
"x3": 4,
"y3": 3,
"x4": 0,
"y4": 3
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/quadrilateral-area", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"x1":0,"y1":0,"x2":4,"y2":0,"x3":4,"y3":3,"x4":0,"y4":3}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"x1":0,"y1":0,"x2":4,"y2":0,"x3":4,"y3":3,"x4":0,"y4":3}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/quadrilateral-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
{
"x1": 0,
"y1": 0,
"x2": 4,
"y2": 0,
"x3": 4,
"y3": 3,
"x4": 0,
"y4": 3
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.quadrilateral_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.
Limits
max_abs_coord | 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 with no computer-algebra system, no random seeds, and 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 every time you call it.
What the shoelace formula is and why vertex order matters
When opposite edges cross, the figure is a self-intersecting bow-tie rather than a simple quadrilateral; the shoelace number may still be nonzero, but it no longer represents a single simple region, so this calculator rejects that case with invalid_input instead of inventing an area.
How to call the API and what comes back
Use it from geometry homework pipelines, cadastral takeoff scripts, game-level tooling, interview-prep sandboxes, room-area checks from wall corners, or any workflow that needs a reproducible four-vertex area without installing a GIS or computer-algebra package on every machine in the path. Send a JSON body with eight numeric fields: x1 and y1 for the first vertex, x2 and y2 for the second, x3 and y3 for the third, and x4 and y4 for the fourth, each in sequential boundary order. Short aliases ax, ay through dx, dy and x_a, y_a through x_d, y_d are accepted for the same coordinates. On success the response echoes the four vertices, returns area and signed_area, lists the four side_lengths and the perimeter, sets orientation to clockwise or counterclockwise, sets method to shoelace, and includes a short formula block documenting the closed form. Failed validation—missing a required coordinate, a non-finite number, a coordinate outside published absolute limits, consecutive or pairwise duplicate vertices, a self-intersecting bow-tie where opposite edges cross, or a degenerate zero-area figure—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
When teaching, prefer paired examples: the unit square for area one; a three-by-four rectangle for area twelve; and a bow-tie such as (0,0), (2,2), (0,2), (2,0) to demonstrate the hard rejection path that protects callers from a self-intersecting figure. Prefer shoelace when the corners are already measured in a plane and you need the enclosed simple area without reconstructing heights. Round-trip checks are easy: after solving, confirm that perimeter equals the sum of the four side lengths and that area matches half the absolute shoelace sum within floating tolerance for the same ordered list. Keep every coordinate absolute value within one trillion so intermediate products stay finite in IEEE double precision on the Worker and in the browser. Units are whatever you pass—metres, feet, or game units—as long as all eight numbers share the same unit; the API does not convert CRS or units for you. Because the formula only needs ordered vertices, students can recompute the eight terms by hand for small integers and match the API line by line. This landing targets people searching for a quadrilateral area calculator or a shoelace formula tool for four corners; it is not a triangle SSS solver—use the dedicated Heron capability when you know three side lengths instead of coordinates. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when users type crossed edges while learning the contract.
Integration use cases
Homework and exam checks for four-vertex area
Give students four ordered corner coordinates; compare their hand-computed shoelace area and orientation against a deterministic API or free browser result.