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 CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published modes
Assurance
Core
Declared partition coverage
PASS · 8/8 declared partitions (rectangle, square, regular-polygon, annulus, parallelogram, trapezoid, irregular-polygon, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Unsigned shoelace area, perimeter, and an engine-linked figure of a simple 3–8 vertex polygon.
Scope
Euclidean plane
Verification
Engine tested · Source checked · v1.5.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.5.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-22.geometry-cvp
Verification revision
2026-09-22.geometry-cvp · 4/4 property · digest dcd9f429cc7c
Legacy regression
25/25 tests · Production surface contract 5/5
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
10/10 golden · 5/5 CVP boundary · 11/11 invalid · 4/4 property · 5/5 cross-interface · 5/5 cross-calculator · 5/5 CVP contract · Manifest
Sources
Sources
Evidence
9 legacy golden · 5 legacy boundary · legacy regression suite · 10/10 oracle-backed golden · 11/11 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.5.0 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

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. The figure updates with the vertices.

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 1.5.0 · 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, perimeter, and an engine-linked figure 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 5/5 · 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
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
    Supports: Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.
  • Gilbert Strang, Calculus — Chapter 15: Vector Calculus (MIT OpenCourseWare) — §15.3 Green’s Theorem
    Supports: The planar area formula A = (1/2) ∮(x dy − y dx). For straight polygon edges it gives the shoelace expression; rectangular/square areas and trapezoid area (B+b)h/2 follow by applying it to the vertices. For x=a cos t, y=b sin t it gives ellipse area πab. Perimeters of these polygons are sums of side lengths. These formulas are derived applications of the theorem. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
Calculation version
1.5.0

Background

Interpretation and common distinctions.

What does this irregular polygon calculator compute?

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.

The published formula or method is A = (1/2)|Σ (xᵢ yᵢ₊₁ − xᵢ₊₁ yᵢ)|. The following references support the stated mathematics or method:

  • Gilbert Strang, Calculus — Chapter 15: Vector Calculus (MIT OpenCourseWare): The planar area formula A = (1/2) ∮(x dy − y dx). For straight polygon edges it gives the shoelace expression; rectangular/square areas and trapezoid area (B+b)h/2 follow by applying it to the vertices. For x=a cos t, y=b sin t it gives ellipse area πab. Perimeters of these polygons are sums of side lengths. These formulas are derived applications of the theorem. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.

Which inputs and limits should you check?

Use one consistent length unit; area uses its square. Select the actual shape and follow the positivity and vertex-order constraints shown in the specification.

The worked example uses 0,0; 5,0; 4,4; 0,3 with A = (1/2)|Σ (xᵢ yᵢ₊₁ − xᵢ₊₁ yᵢ)|. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.geometry.plane v1.5.0 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Other calculators in this family: Annulus Calculator, Arc Calculator, Circle Calculator, Conic Section Calculator, Ellipse Calculator, Hyperbola Calculator, Parabola Calculator, Parallelogram Calculator . Explore all Geometry.

→ Annulus Calculator Annulus A = π(R² − r²) and P = 2π(R + r) with outer R > inner r. Same engine as Plane Geometry. Not /circle. Not a square. Not CAS. Runs locally. › → Arc Calculator Calculate circular arc length from radius and central angle in degrees. See the radian formula, worked examples, units, and mathematical sources. › → Circle Calculator Calculate circle radius, diameter, circumference, area, arc length, and sector area. Explore formulas, worked examples, units, and mathematical sources. › → 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. › → Ellipse Calculator Standard ellipse x²/a² + y²/b² = 1. Same engine as Conic Section. a = b is a valid ellipse with e = 0; the circle-only page is /circle. Not CAS. Runs locally. › → Hyperbola Calculator Standard hyperbola x²/a² − y²/b² = 1 (horizontal) or y²/a² − x²/b² = 1 (vertical). Same engine as Conic Section. Not CAS. Runs locally. › → Parabola Calculator Standard parabola x² = 4py (vertical) or y² = 4px (horizontal). Same engine as Conic Section. p ≠ 0. Not CAS. Runs locally. › → Parallelogram Calculator Parallelogram A = ab sin θ, P = 2(a+b), diagonals from a, b, and included angle. Same engine as Plane Geometry. Not a rectangle page. Not the fluid trapezoid. Not CAS. Runs locally. ›

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.