Calculate the area, perimeter, centroid, and bounding box of any irregular polygon using the shoelace formula. Enter up to 10 vertex coordinates and see step-by-step cross products.
The shoelace formula (also known as the surveyor's formula or Gauss's area formula) computes the area of any simple polygon when you know its vertex coordinates. It works by summing cross products of consecutive vertex pairs — a method that is both elegant and computationally efficient.
For a polygon with vertices (x₁,y₁), (x₂,y₂), …, (xₙ,yₙ) listed in order, the signed area is A = ½ × Σᵢ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ), where indices wrap around (vertex n+1 = vertex 1). The absolute value gives the true area regardless of vertex ordering direction.
Beyond area, the same vertex data yields the perimeter (sum of edge lengths via the distance formula), the centroid (the polygon's geometric center, computed from a weighted average of vertex cross products), and the bounding box (the smallest axis-aligned rectangle enclosing the polygon).
This calculator accepts up to 10 vertex coordinates and computes all four quantities. It shows the shoelace cross products step by step — ideal for students learning coordinate geometry. Presets include a square, right triangle, L-shape, hexagonal shape, arrow, and plus sign. Edge lengths are displayed with proportional bars, and the bounding box fill ratio tells you how "rectangular" the polygon is.
The shoelace formula requires a simple (non-self-intersecting) polygon. If edges cross, the result will be incorrect. Vertices must be entered in consecutive order (clockwise or counter-clockwise).
The Irregular Polygon Area Calculator — Shoelace Formula is useful when you need fast and consistent geometry results without reworking the same algebra repeatedly. It helps you move from raw measurements to Area, Perimeter, Centroid in one pass, with conversions and derived values shown together.
Use it to validate homework steps, check CAD or fabrication dimensions, estimate material requirements, and sanity-check hand calculations before submitting work.
Shoelace Area: A = ½|Σᵢ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)| Perimeter: P = Σᵢ √((xᵢ₊₁ − xᵢ)² + (yᵢ₊₁ − yᵢ)²) Centroid: Cx = (1/6A) × Σᵢ(xᵢ + xᵢ₊₁)(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ) Cy = (1/6A) × Σᵢ(yᵢ + yᵢ₊₁)(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ) Bounding Box: min/max of all x and y coordinates
Result: Area = 12, Perimeter ≈ 16, Centroid ≈ (1.333, 1.333)
For an L-shape with 6 vertices: cross products sum to 24, so area = ½ × |24| = 12. The edges have lengths 4, 2, 2, 2, 2, 4 → perimeter = 16. The centroid is weighted toward the larger part of the L.
This calculator combines the core geometry formula with the input mode selected in the interface, then derives companion values shown in the output cards, comparison bars, and reference tables. Use it to cross-check both direct calculations and reverse-solving scenarios where one measurement is unknown.
Irregular Polygon Area Calculator — Shoelace Formula calculations show up in coursework, drafting, construction layout, packaging, tank sizing, machining, and quality control. Instead of solving each transformation manually, you can test scenarios quickly and verify whether your dimensions remain within tolerance.
Keep units consistent across every input before interpreting area, perimeter, angle, or volume outputs. For best results, measure carefully, round only at the final step, and compare at least one manual calculation with the calculator output when building confidence.
A mathematical formula that computes the area of a simple polygon from its vertex coordinates: A = ½|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|. It's called "shoelace" because the cross-multiplication pattern resembles lacing.
The signed area changes sign with vertex order (clockwise vs. counter-clockwise), but the absolute area is the same. The calculator uses the absolute value.
Yes! The shoelace formula works for any simple polygon, including concave ones. Just ensure edges don't cross each other.
The formula gives incorrect results for self-intersecting polygons. Split them into non-overlapping simple polygons and compute each separately.
For a polygon with uniform density, the centroid formula is exact. It computes the true center of mass of the planar region.
It's the polygon area divided by its axis-aligned bounding box area. A perfect rectangle has 100%. It measures how "efficiently" the polygon fills its bounding rectangle.
For small areas (a few km²), you can use projected coordinates (e.g., UTM). For large areas, use a geodesic formula instead, as the curvature of the Earth matters.