Try an Example
Pick a scenario to see how the calculator works, then adjust the values
Signal Addition
Add two phasor signals in complex form to find the resultant.
Key values: z₁ = 3 + 4i · z₂ = 1 − 2i · Addition
Circuit Impedance
Divide two impedances to compute the transfer ratio.
Key values: z₁ = 3 + 4i · z₂ = 1 + 2i · Division
De Moivre Power
Raise a complex number to the 10th power using De Moivre's theorem.
Key values: z₁ = 1 + i · n = 10 · Power
Cube Roots of −8
Find all three cube roots of −8, visualized as equidistant points on a circle.
Key values: z₁ = −8 + 0i · n = 3 · nth Root
This calculator is also known as Rectangular to Polar Form Calculator.
Read the complete guideStep-by-Step Conversion
Given z = a + bi: (1) Compute r = √(a² + b²). (2) Compute θ = atan2(b, a) in radians. (3) Convert to degrees if needed: θ° = θ × 180/π. (4) Write in polar form: z = r∠θ° or z = r·cis(θ).
Examples
Convert −1 − √3·i to Polar Form
Find the polar form of z = −1 − √3·i.
r = √(1 + 3) = 2. θ = atan2(−√3, −1) = −2π/3 = −120°. So z = 2∠(−120°).
Key takeaway: Points in the third quadrant have negative arguments (below the real axis). The atan2 function handles this automatically.
Rectangular-to-Polar Tips
Common tips for form conversion:
- Memorize common values: |1+i| = √2, arg(1+i) = 45°; |1+√3·i| = 2, arg = 60°
- Use atan2 to avoid quadrant errors — it is built into every scientific calculator and programming language
- In exam settings, sketch the point on the Argand diagram first to verify the quadrant visually
Frequently Asked Questions about Rectangular to Polar Form Calculator
Why not use arctan(b/a) instead of atan2(b, a)?
arctan(b/a) only gives angles in (−π/2, π/2), which covers only the right half of the plane. atan2(b, a) returns the correct angle in (−π, π] for all four quadrants and handles a = 0.
How do I convert back to rectangular form?
Given r∠θ: a = r·cos(θ) and b = r·sin(θ). So z = r·cosθ + i·r·sinθ.
Specialized Calculators
Choose from 11 specialized versions of this calculator, each optimized for specific use cases and calculation methods.
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