Skip to main content

Polygon Area Calculator

Geometry

Shoelace formula from vertex coordinates, centroid

Quick Answer

Polygon Area Calculator: Shoelace formula from vertex coordinates, centroid. Polygon Area Calculator uses standard geometry formulas to compute accurate results instantly.

5 vertices parsed (need ≥ 3)

Polygon area (shoelace)82.0000
Perimeter35.3710
OrientationCounter-clockwise
Centroid X5.935
Centroid Y4.033
centroid

Shoelace steps

  1. 15 vertices in boundary order; pair each vertex with the next (last wraps to first)
  2. 2Cross-sum Σ(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ) = 164.0000
  3. 3Sign → orientation: positive = counter-clockwise
  4. 4Area = |Σ| / 2 = 164.0000 / 2 = 82.0000
  5. 5Centroid = weighted average of edge midpoints by cross product → (5.935, 4.033)

Vertices

#xyEdge to next
10.00000.000010.0000
210.00000.00005.3852
312.00005.00005.6569
48.00009.00006.0828
52.00008.00008.2462
Shoelace formula: A = ½|Σ(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)| — works for ANY simple polygon (convex or concave). Used in GIS land surveying, CAD, CNC nesting. Vertices must be listed in boundary order, not sorted by angle.

Was this tool helpful?

How to Use

  1. Enter known values: fill sides, radius or angles in the Polygon Area Calculator.
  2. Auto-solve the rest: all unknowns are solved instantly — no manual derivation.
  3. See formula and steps: the result panel shows the formula used and the substitution.
  4. Try another case: change any known value to solve again.

Key Points

  • Polygon Area Calculator is a free online geometry tool — open and use, no sign-up.
  • All calculations run locally in your browser; your data never leaves the device.
  • Key capability: Shoelace formula from vertex coordinates, centroid.
  • Results come with formulas and reference tables so you can verify every number.

Formula & How It Works

Polygon Area Calculator formula

Polygon Area Calculator uses standard geometry formulas to compute accurate results instantly.

FAQ

How do I use the Polygon Area Calculator?
Open the page and fill in the input fields — results update automatically. Shoelace formula from vertex coordinates, centroid.
Are the results accurate?
Yes. Standard formulas and conversion factors are used and verified.
Is it free? Do I need to sign up?
Completely free. No registration, login or download required.
Does it work on mobile?
Yes. The responsive design works on phones, tablets and desktops.
Is my data saved?
No. Everything runs locally in your browser.

Related Tools

Related Guides

Source & Updates

Based on standard Euclidean geometry formulas. Last updated: 2025-07

Last updated:

Formulas and content on this page are reviewed by the YiJianSuan team. For reference only — official sources prevail.

Maintained and periodically reviewed by the YiJianSuan editorial team — review process in ourEditorial Policy.

Cite this page: YiJianSuan. Polygon Area Calculator[EB/OL]. https://www.yijiansuan.com/en/c/geometry/shoelace

Results provided by this tool are for reference only and do not constitute professional advice.