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Polygon Area Visualizer — Shoelace Formula

Animated polygon area with the shoelace formula — sums signed triangle areas with step controls. Runs in your browser.

Free No signup Client-side Privacy friendly Updated

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Pseudocode

Press Run to animate the algorithm.

How to use

  1. 1 Press Run to compute the polygon’s area with the shoelace formula.
  2. 2 Watch each triangle fanned from the first vertex add to the running total.
  3. 3 Use Shuffle for a new random polygon, or step through one triangle at a time.
  4. 4 The final value is the polygon’s area in square units.

Why use this tool

  • See the shoelace (Gauss’s area) formula build the area triangle by triangle.
  • Understand how signed cross products combine into the total area.
  • Works for any simple polygon — convex or concave — in O(n).
  • Runs entirely in your browser. No signup, no uploads.

Frequently asked questions

What is the shoelace formula?

A formula for the area of a simple polygon from its vertex coordinates: sum the cross products of consecutive vertices and halve the absolute value. It is named for the criss-cross multiplication pattern.

What is the time complexity?

O(n) — a single pass over the polygon’s n vertices, with O(1) extra space.

Does it work for concave polygons?

Yes, as long as the polygon is simple (its edges do not cross). The signed triangle areas correctly cancel for concave regions.

Why divide by two?

Each cross product equals twice the signed area of the triangle it represents, so the total sum is twice the polygon’s area.

What is Polygon Area Visualizer (Shoelace)?

A Polygon Area Visualizer animates the shoelace (Gauss) formula: it fans triangles from the first vertex and sums their signed cross-product areas to compute the area of any simple polygon in O(n) time.

Features

Shoelace computation

Visualize the cross-product terms that sum to twice the area.

Complexity

Time: O(n). Space: O(1). Works for any simple (non-self-intersecting) polygon.

100% private

Runs entirely in your browser — nothing is uploaded.

Example

Input

triangle (0,0), (4,0), (0,3)

Output

area = 6

Common use cases

  1. 1

    Land & map areas

    Compute the area of a plotted region from its corner coordinates.

  2. 2

    Graphics & CAD

    Get polygon areas for rendering, physics, or design.

  3. 3

    Learn the shoelace formula

    See why the signed cross-product sum gives the area.

Summary

Zerethon's polygon area visualizer computes the area of a simple polygon from its vertices in your browser using the shoelace formula (Gauss's area formula), which sums cross products of consecutive vertices. It runs in O(n) time and O(1) extra space for an n-vertex polygon.

Category
Algorithm
Pricing
Free
Privacy
Browser-based
Signup
Not required

References

Privacy

Your data never leaves your browser unless explicitly stated. Polygon Area Visualizer (Shoelace) runs entirely client-side — no server upload, no logging, no tracking of your input.

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