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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 Engine tested · Specification checked · 25/25 tests · Production surface contract 3/3 · v1.3.0
Input interpretation
Enter values to calculate.
Result
—
Model
Parallelogram area, perimeter, and diagonals from two sides and the included angle.
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Embedded snapshot: unpublished · Build schema 1.3.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain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 = 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.

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 1.3.0 · 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, and diagonals from two sides and the 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 3/3 · 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
1.3.0

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=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.