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Math calculator

Complex Polar Calculator

Convert a complex number between rectangular (re, im) and polar (r, θ°) form. Same engine as Complex Number. Not a CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 18/18 tests · Production surface contract 3/3 · v1.1.0
Input interpretation
Enter values to calculate.
Result
—
Model
Rectangular ↔ polar conversion of one complex number.
Scope
atan2 argument in degrees
Verification
Engine tested · 18/18 tests · Production surface contract 3/3 · Specification checked · v1.1.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
6 golden · 4 boundary · 8 property · Production surface contract 3/3 · Artifact integrity PASS
Production
Embedded snapshot: STALE · Last attested schema matched 1.1.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓ · Live production status STALE (1 capability; 163 remain CURRENT) @ 2026-09-19T00:00:17.039Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

Modulusr = √(re² + im²)
Argumentθ = atan2(im, re)
Rectangularre = r cos θ, im = r sin θ
iθ in degrees, atan2 range (−180°, 180°]. r ≥ 0. Not symbolic simplify and not a second complex engine.

How to use

1

Choose direction

Rect → polar needs re and im. Polar → rect needs r and θ in degrees.

2

Read both forms

Every result includes re, im, r, and θ. Arithmetic, conjugate, and modulus remain on Complex Number.

Example calculations

Common configurations with formula and result.

ϟ

3+4i

rect to polar

r=5
5 ∠ 53.13°
ϟ

5 ∠ 53.13°

polar to rect

3+4i
3 + 4i

Complex Polar calculator specification

Version 1.1.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
z = re + im i = r (cos θ + i sin θ) with r = |z| and θ = atan2(im, re) in degrees. This page is a discovery surface on the complex engine (rect_to_polar / polar_to_rect). Arithmetic stays on /calc/math/complex-number.
What it calculates
Rectangular ↔ polar conversion of one complex number.
Inputs
  • mode
  • re?
  • im?
  • r?
  • arg_deg?
Outputs
  • re
  • im
  • r
  • arg_deg
Formula
re = r cos θ, im = r sin θ
Assumptions
  • atan2 argument in degrees
  • r ≥ 0
  • Not a CAS
Units
  • dimensionless
  • degrees
Boundary conditions
  • r < 0 → VALUE_MUST_BE_NON_NEGATIVE
  • unknown mode (including simplify) → INVALID_MODE
Example
mode=rect_to_polar re=3 im=4 → r=5
Validation cases

2 published on this page · 18/18 tests · Production surface contract 3/3 · View evidence

  • mode=rect_to_polar re=3 im=4 → r=5
  • mode=polar_to_rect r=5 arg_deg=53.1301023542 → re≈3 im≈4
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.1.0

Other calculators in this family: Complex Number Calculator, Complex Power Calculator, Complex Roots Calculator, Euler Form Calculator . Explore all Complex Numbers.

→ Complex Number Calculator Add, subtract, multiply, divide complex numbers; convert rectangular ↔ polar; integer powers, nth roots, and Euler form. Not a CAS. Runs locally. › → Complex Power Calculator Integer power of a complex number via De Moivre. Same engine as Complex Number. Not a CAS. Not i^i. Runs locally. › → Complex Roots Calculator All n nth-roots of a complex number via De Moivre. Same engine as Complex Number. Not a CAS. Not the real root page. Runs locally. › → Euler Form Calculator Euler form of a complex number: z = r e^{iθ} = r (cos θ + i sin θ). Same engine as Complex Number. Not a CAS. Runs locally. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this a second complex engine?

No. It is rect_to_polar / polar_to_rect on math.complex, the same engine as /calc/math/complex-number.

Can I add two complex numbers here?

Use /calc/math/complex-number for add, subtract, multiply, divide, conjugate, and modulus.