HomeCalculatorsMathGeometryIrregular Polygon Calculator
Math calculator

Irregular Polygon Calculator

Shoelace area of a simple 3–8 vertex polygon. Same engine as Plane Geometry. Not /regular-polygon. Not a triangle solver. Not CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 25/25 tests · Production surface contract 3/3 · v1.3.0
Input interpretation
Enter values to calculate.
Result
—
Model
Unsigned shoelace area and perimeter of a simple 3–8 vertex polygon.
Scope
Euclidean plane
Verification
Engine tested · 25/25 tests · Production surface contract 3/3 · Specification checked · v1.3.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
9 golden · 5 boundary · 11 property · Production surface contract 3/3 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.3.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain CURRENT) @ 2026-09-20T09:01:12.147Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

AreaA = (1/2)|Σ (xᵢ yᵢ₊₁ − xᵢ₊₁ yᵢ)|
PerimeterP = Σ √((xᵢ₊₁ − xᵢ)² + (yᵢ₊₁ − yᵢ)²)
iSame engine as /plane-geometry. Regular n-gons stay on /regular-polygon. SSS/SAS stay on /triangle-solver. Not a second engine.

How to use

1

Enter 3–8 vertices

Use x,y; x,y pairs. Default 0,0; 5,0; 4,4; 0,3. Consecutive vertices must be distinct. A line or collinear set is invalid.

2

Read A and P

The default quadrilateral has area 16. Rectangle vertices (0,0),(3,0),(3,4),(0,4) recover A=12, P=14.

Example calculations

Common configurations with formula and result.

ϟ

Default quadrilateral

0,0; 5,0; 4,4; 0,3

A = (1/2)|Σ (xᵢ yᵢ₊₁ − xᵢ₊₁ yᵢ)|
A=16
ϟ

Rectangle identity

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

A = 12, P = 14
A=12, P=14

Irregular Polygon calculator specification

Version 1.3.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A simple polygon with 3–8 vertices (xᵢ, yᵢ) has unsigned shoelace area (1/2)|Σ(xᵢ yᵢ₊₁ − xᵢ₊₁ yᵢ)| and perimeter Σ√((xᵢ₊₁−xᵢ)²+(yᵢ₊₁−yᵢ)²). Closing duplicate vertices are dropped. This page is mode=irregular_polygon on math.geometry.plane. Not /regular-polygon. Not a triangle solver. Not CAS.
What it calculates
Unsigned shoelace area and perimeter of a simple 3–8 vertex polygon.
Inputs
  • vertices
Outputs
  • area
  • perimeter
  • n
Formula
A = (1/2)|Σ (xᵢ yᵢ₊₁ − xᵢ₊₁ yᵢ)|
Assumptions
  • Euclidean plane
  • Simple polygon, unsigned |A|
  • Closing duplicate vertex dropped
  • Not /regular-polygon
  • Not a triangle solver
  • Not CAS
Units
  • length (consistent units)
  • area (length²)
Boundary conditions
  • n < 3 or n > 8 → INVALID_INPUT
  • consecutive duplicates → INVALID_INPUT
  • collinear or A=0 → INVALID_INPUT
  • unknown mode → INVALID_MODE
Example
vertices 0,0; 5,0; 4,4; 0,3 → A=16
Validation cases

3 published on this page · 25/25 tests · Production surface contract 3/3 · View evidence

  • vertices=0,0; 5,0; 4,4; 0,3 → A=16 n=4
  • vertices=(0,0),(3,0),(3,4),(0,4) → A=12 P=14
  • vertices=0,0; 1,0 → INVALID_INPUT
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.3.0

Other calculators in this family: Angle Between Lines Calculator, Circle Calculator, Cone Calculator, Conic Section Calculator, Coordinate Geometry Calculator, Cross Product Calculator, Cylinder Calculator, Dot Product Calculator . Explore all Geometry.

→ Angle Between Lines Calculator Acute angle between two plane lines, each given by two points. Parallel is 0°. Perpendicular is 90°. Degenerate points are invalid. Same engine as Coordinate Geometry. Not a second engine. Not CAS. Runs locally. › → Circle Calculator Compute diameter, circumference, and area from r, or arc and sector from a central angle. Not the Pythagorean page. Runs locally. › → Cone Calculator Right circular cone V = (1/3)πr²h and slant l = √(r²+h²). Same engine as Solid Geometry. Not ISO 5167-5. Not CAS. Runs locally. › → Conic Section Calculator Axis-aligned ellipse, parabola, and hyperbola in standard form. Discovery /ellipse, /parabola, /hyperbola. Not the circle-only page. Not coordinate geometry. Not CAS. Runs locally. › → Coordinate Geometry Calculator Distance, midpoint, and slope of two points in the plane. Line through two points is /line-equation. Point-to-line distance is /point-line-distance. Intersection is /line-intersection. Angle is /angle-between-lines. Not the Pythagorean-only page. Runs locally. › → Cross Product Calculator Vector product u × v in R³. 2-D uses uz = vz = 0. Same engine as Vector. Not a matrix. Not CAS. Runs locally. › → Cylinder Calculator Right circular cylinder V = πr²h and A = 2πr(r+h). Same engine as Solid Geometry. Not CAS. Runs locally. › → Dot Product Calculator Scalar product u · v = ux vx + uy vy + uz vz. Same engine as Vector. Not a matrix. Not CAS. Runs locally. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this a second plane engine?

No. It is mode=irregular_polygon on math.geometry.plane, the same engine as /calc/math/plane-geometry.

Is this the regular polygon page?

No. Equal-side n-gons stay on /calc/math/regular-polygon. The alias polygon still maps to regular_polygon on the hub.

Is this a triangle solver?

No. SSS/SAS stay on /calc/math/triangle-solver. A three-vertex shoelace is allowed here, but it is not the default.

Where does this run?

Locally in the browser by default. REST and MCP call the same plane-geometry engine.