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Updated 2026-03-01 UTC
Rectangular to Polar — Complex Magnitude & Angle
Convert complex numbers from rectangular to polar form with GetCalcMaster. Magnitude/angle workflow with safe checks.
Use this guide to convert a+jb into r∠θ, compute magnitude and angle, and verify results with back-conversion.
Important: Educational use only. Angle conventions vary (−π..π vs 0..2π). Confirm the convention you need.
What this calculator is
The Complex Number Calculator is an interactive tool inside GetCalcMaster. It’s designed to help you explore scenarios, understand formulas, and document assumptions.
Key features
- Magnitude: r=√(a²+b²)
- Angle: θ=atan2(b,a)
- Verify by converting back to a+jb
Formula
Given a + bi: r = √(a² + b²)
θ = atan2(b, a)
Result: r∠θQuick examples
3 + 4i → r=5, θ≈53.13°−1 + 0i → r=1, θ=180°0 − 2i → r=2, θ=−90° (or 270°)
Verification tips
- Use atan2(b, a) to get the correct quadrant.
- Sanity check: converting back should reproduce a + bi.
- θ can be expressed in degrees or radians—stay consistent.
Common mistakes
- Using atan(b/a) (loses quadrant information).
- Dropping the sign of b (imaginary part).
- Forgetting that multiple angles represent the same direction (θ + 2πk).
How to use it (quick steps)
- Enter complex numbers in the supported form (a+bi, polar, etc.).
- Choose the operation (add/multiply, magnitude/phase, conversions).
- Review both rectangular and polar interpretations where relevant.
- Copy/export results or send them to Notebook for future reference.
Related tools and guides
Featured guides
Deep, human-written guides focused on accuracy, verification, and reproducible workflows.
Complex number guide
Rectangular and polar workflows.
Polar → rectangular
Convert r∠θ to a+jb.
Rectangular → polar
Magnitude/angle workflow.
Math hub
Tools + explainers across math topics.
FAQ
Why does my angle differ by 360° (or 2π)?
Angles are periodic. θ and θ+2π represent the same direction; choose the convention required by your context.
What is atan2 and why is it better than atan?
atan2 uses both a and b to pick the correct quadrant, avoiding sign/quadrant mistakes.
Tip: For reproducible work, save your inputs and reasoning in Notebook.