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Math calculator

Plane Geometry Calculator

Rectangle, square, regular n-gon, annulus, parallelogram, isosceles trapezoid, and irregular polygon (shoelace). Discovery /rectangle, /square, /regular-polygon, /annulus, /parallelogram, /trapezoid, /irregular-polygon. Not the circle-only page. Not a triangle solver. Not solid geometry. Not /square-root. 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
Rectangle, square, regular n-gon, annulus, parallelogram, isosceles trapezoid, and irregular polygon area, perimeter, and an engine-linked figure per mode.
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
  • ISO 80000-2:2019 Quantities and units — Mathematics
  • CRC Standard Mathematical Tables — plane polygons
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.

RectangleA = ab, P = 2(a+b), d = √(a²+b²)
Regular polygonA = (n s²) / (4 tan(π/n))
AnnulusA = π(R² − r²)
ParallelogramA = ab sin θ
TrapezoidA = (a+b)h/2
Irregular polygonA = (1/2)|Σ (xᵢ yᵢ₊₁ − xᵢ₊₁ yᵢ)|
iSquare is /calc/math/square. Discovery pages lock rectangle, square, regular polygon, annulus, parallelogram, trapezoid, and irregular polygon. Not /square-root. Not CAS.

How to use

1

Choose rectangle, square, regular polygon, annulus, parallelogram, trapezoid, or irregular polygon

Rectangle needs a>0 and b>0. Regular polygon needs integer n≥3 and side s>0. Annulus needs outer R greater than inner r. Parallelogram needs a, b, and θ∈(0°,180°). Isosceles trapezoid needs bases a, b and height h>0. Irregular polygon needs 3–8 vertices as x,y; x,y.

2

Read area and perimeter

A 3×4 rectangle has diagonal 5. A regular hexagon with s=1 has interior 120°. The figure updates with the selected mode.

Example calculations

Common configurations with formula and result.

ϟ

3-4 rectangle

a = 3, b = 4

d = √(9+16)
A=12, d=5
ϟ

Hexagon

n = 6, s = 1

A = 3√3/2
P=6, 120°
ϟ

Annulus

R = 2, r = 1

A = 3π
3π
ϟ

Parallelogram 2-3-30°

a = 2, b = 3, θ = 30°

A = 6 sin 30°
A=3, P=10
ϟ

Trapezoid 2-4-3

a = 2, b = 4, h = 3

A = (2+4)×3/2
A=9
ϟ

Irregular quadrilateral

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

A = (1/2)|Σ (xᵢ yᵢ₊₁ − xᵢ₊₁ yᵢ)|
A=16

Plane Geometry calculator specification

Version 1.5.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Euclidean plane figures: rectangle A=ab with diagonal √(a²+b²), square A=s², regular n-gon A=(n s²)/(4 tan(π/n)) for integer n≥3, annulus A=π(R²−r²) with R>r, parallelogram A=ab sin θ for θ∈(0°,180°), isosceles trapezoid A=(a+b)h/2, irregular polygon shoelace A=(1/2)|Σ(xᵢyᵢ₊₁−xᵢ₊₁yᵢ)| on 3–8 vertices. Not the circle-only page. Not a triangle solver. Not /trapezoidal-rule. Not the fluid trapezoid. Not CAS.
What it calculates
Rectangle, square, regular n-gon, annulus, parallelogram, isosceles trapezoid, and irregular polygon area, perimeter, and an engine-linked figure per mode.
Inputs
  • mode
  • a?
  • b?
  • s?
  • n?
  • r?
  • R?
  • theta?
  • h?
  • vertices?
Outputs
  • area
  • perimeter
  • diagonal?
  • d1?
  • d2?
  • height?
  • leg?
  • midsegment?
  • apothem?
  • interior_deg?
Formula
A = ab; A = (n s²)/(4 tan(π/n)); A = π(R² − r²); A = ab sin θ; A = (a+b)h/2; A = (1/2)|Σ(xᵢyᵢ₊₁−xᵢ₊₁yᵢ)|
Assumptions
  • Euclidean plane
  • Regular polygon integer n ≥ 3
  • Annulus R > r
  • Parallelogram θ in (0°, 180°)
  • Isosceles trapezoid a, b, h > 0
  • Irregular polygon shoelace 3–8 vertices
  • Not the circle-only page
  • Not a triangle solver
  • Not solid geometry
  • Not /trapezoidal-rule
  • Not the fluid trapezoid
  • Not CAS
Units
  • length (consistent units)
  • area (length²)
  • degrees
Boundary conditions
  • a ≤ 0 or b ≤ 0 or s ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE
  • n < 3 or non-integer n → INVALID_INPUT
  • annulus R ≤ r → INVALID_INPUT
  • parallelogram θ ≤ 0° or θ ≥ 180° → INVALID_INPUT
  • irregular n < 3 or n > 8 or A=0 → INVALID_INPUT
  • unknown mode → INVALID_MODE
Example
rectangle a=3 b=4 → A=12, d=5
Validation cases

7 published on this page · 25/25 tests · Production surface contract 5/5 · View evidence

  • mode=rectangle a=3 b=4 → A=12 P=14 d=5
  • mode=square s=2 → A=4 P=8 d=2√2
  • mode=regular_polygon n=6 s=1 → P=6 interior=120°
  • mode=annulus R=2 r=1 → A=3π
  • mode=parallelogram a=2 b=3 theta=30 → A=3 P=10
  • mode=trapezoid a=2 b=4 h=3 → A=9
  • mode=irregular_polygon vertices=0,0; 5,0; 4,4; 0,3 → A=16 n=4
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
    Supports: Quantity names and symbols for plane area and perimeter
  • CRC Standard Mathematical Tables — plane polygons
    Supports: Regular n-gon A = (n s²)/(4 tan(π/n)); parallelogram A = ab sin θ; shoelace / Meister–Gauss polygon area
Calculation version
1.5.0

Other calculators in this family: Annulus Calculator, Arc Calculator, Circle Calculator, Conic Section Calculator, Ellipse Calculator, Hyperbola Calculator, Irregular Polygon Calculator, Parabola 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 Circle arc s = r θ from radius and central angle. Same engine as Circle. Not /sector. Not /spherical-sector. Not CAS. Runs locally. › → Circle Calculator Compute diameter, circumference, and area from r, or arc and sector from a central angle. Discovery /sector, /arc. Not the Pythagorean page. Not /spherical-sector. 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. › → 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. › → 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. › → Parabola Calculator Standard parabola x² = 4py (vertical) or y² = 4px (horizontal). Same engine as Conic Section. p ≠ 0. Not CAS. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Is the 3-4 rectangle the Pythagorean calculator?

No. The diagonal is the same 5 as a 3-4-5 triangle, on a different model. Right triangles stay on /calc/math/pythagorean.

Is this the circle calculator?

No. Diameter, circumference, arc, and sector stay on /calc/math/circle. An annulus with R=r is invalid here.

Where does this run?

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