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Math calculator

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.

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
Parallelogram area, perimeter, diagonals, and an engine-linked figure of the sides and included angle.
Scope
Euclidean plane
Verification
Engine tested · Specification 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
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
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 = ab sin θ
Diagonalsd² = a² + b² ± 2ab cos θ
iSame engine as /plane-geometry. A rectangle is mode=rectangle on that hub, or θ=90° here. Not a second engine.

How to use

1

Enter a, b, and θ

Sides must be positive. Included angle is in degrees, strictly between 0° and 180°. Default a=2, b=3, θ=30°.

2

Read A, P, and the diagonals

2-3-30° has area 3 and perimeter 10. 3-4-90° matches the 3-4 rectangle. The figure updates with a, b, and θ.

Example calculations

Common configurations with formula and result.

ϟ

2-3-30°

a = 2, b = 3, θ = 30°

A = 6 sin 30°
A=3, P=10
ϟ

Rectangle identity

a = 3, b = 4, θ = 90°

A = 12, d1=d2=5
A=12, d=5

Parallelogram calculator specification

Version 1.5.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A parallelogram of sides a>0, b>0 and included angle θ∈(0°,180°) has area ab sin θ, perimeter 2(a+b), and diagonals √(a²+b²±2ab cos θ). θ=90° recovers a rectangle. This page is mode=parallelogram on math.geometry.plane. Not a second engine. Not the fluid trapezoid. Not CAS.
What it calculates
Parallelogram area, perimeter, diagonals, and an engine-linked figure of the sides and included angle.
Inputs
  • a
  • b
  • theta
Outputs
  • area
  • perimeter
  • d1
  • d2
  • height
Formula
A = ab sin θ, P = 2(a+b), d² = a²+b²±2ab cos θ
Assumptions
  • Euclidean plane
  • θ in (0°, 180°)
  • θ=90° is a rectangle
  • Not the fluid trapezoid
  • Not CAS
Units
  • length (consistent units)
  • area (length²)
  • degrees
Boundary conditions
  • a ≤ 0 or b ≤ 0 → VALUE_MUST_BE_POSITIVE
  • θ ≤ 0° or θ ≥ 180° → INVALID_INPUT
  • unknown mode → INVALID_MODE
Example
a=2 b=3 θ=30 → A=3, P=10
Validation cases

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

  • a=2 b=3 theta=30 → A=3 P=10
  • a=3 b=4 theta=90 → A=12 d1=5 d2=5
  • a=2 b=3 theta=0 → INVALID_INPUT
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
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 this a second plane engine?

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

Is this the fluid trapezoid?

No. Open-channel trapezoid depth stays on /calc/physics. The plane trapezoid is /calc/math/trapezoid on this engine.

Where does this run?

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