Geometry Intermediate

Coordinate Polygon Area — Shoelace Formula

Enter the (x, y) coordinates of 3 to 6 vertices and compute area, perimeter, and convexity using the shoelace formula — with every term shown step by step.

3–6 vertices
Area & perimeter
Convexity check
Input
Try:
Result
Enter vertex coordinates.
Step-by-Step Solution
Polygon Diagram
Shoelace Formula
A = ½|Σ(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)| where indices wrap: after vertex n comes vertex 1 Perimeter = Σ √((xᵢ₊₁−xᵢ)²+(yᵢ₊₁−yᵢ)²)
Convexity

A polygon is convex if all cross products of consecutive edge vectors have the same sign. If any have opposite sign, the polygon is concave (has a "dent").

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